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MATRIXx
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Xmath Xµ Manual
MATRIXx Xmath Basics
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April 2004 Edition
Part Number 370760B-01
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Contents
1
Introduction
1
1
2
3
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Manual Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 How to avoid really reading this Manual . . . . . . . . . . . . . . . . . . .
2
Overview of the Underlying Theory
5
5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 An Introduction to Norms . . . . . . . . . . . . . . . . . . . . . . .
2.2 Modeling Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Perturbation Models for Robust Control . . . . . . . . . . . . . . .
2.2.2 Linear Fractional Transformations . . . . . . . . . . . . . . . . . .
2.2.3 Assumptions on P, ∆, and the unknown signals . . . . . . . . . .
2.2.4 Additional Perturbation Structures . . . . . . . . . . . . . . . . . .
6
8
13
13
17
22
23
iii
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2.2.5 Obtaining Robust Control Models for Physical Systems . . . . . .
2.3 H∞ and H2 Design Methodologies . . . . . . . . . . . . . . . . . . . . . .
2.3.1 H∞ Design Overview . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Assumptions for the H∞ Design Problem . . . . . . . . . . . . . .
2.3.3 A Brief Review of the Algebraic Riccati Equation . . . . . . . . . .
2.3.4 Solving the H∞ Design Problem for a Special Case . . . . . . . . .
2.3.5 Further Notes on the H∞ Design Algorithm . . . . . . . . . . . . .
2.3.6 H2 Design Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.7 Details of the H2 Design Procedure . . . . . . . . . . . . . . . . .
2.4 µ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Measures of Performance . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Robust Stability and µ . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Robust Performance . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Properties of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 The Main Loop Theorem . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 State-space Robustness Analysis Tests . . . . . . . . . . . . . . . .
2.4.7 Analysis with both Real and Complex Perturbations . . . . . . . .
2.5 µ Synthesis and D-K Iteration . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 µ-Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 The D-K Iteration Algorithm . . . . . . . . . . . . . . . . . . . . .
28
29
31
32
33
36
38
40
40
42
42
44
46
47
49
51
58
58
58
60
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2.6 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Truncation and Residualization . . . . . . . . . . . . . . . . . . . .
2.6.2 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Hankel Norm Approximation . . . . . . . . . . . . . . . . . . . . .
64
65
65
68
3
Functional Description of Xµ
71
71
71
72
74
77
78
81
81
81
82
82
85
85
85
88
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Data Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 pdms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Subblocks: selecting input & outputs . . . . . . . . . . . . . . . . .
3.2.4 Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Continuous to Discrete Transformations . . . . . . . . . . . . . . .
3.3 Matrix Information, Display and Plotting . . . . . . . . . . . . . . . . . .
3.3.1 Information Functions for Data Objects . . . . . . . . . . . . . . .
3.3.2 Formatted Display Functions . . . . . . . . . . . . . . . . . . . . .
3.3.3 Plotting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 System Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Creating Time Domain Signals . . . . . . . . . . . . . . . . . . . .
3.4.2 Dynamic System Time Responses . . . . . . . . . . . . . . . . .
3.4.3 Frequency Responses . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5 System Interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 H2 and H∞ Analysis and Synthesis . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . .
91
95
95
3.6.2 System Norm Calculations . . . . . . . . . . . . . . . . . . . . . . 105
3.7 Structured Singular Value (µ) Analysis and Synthesis . . . . . . . . . . . 107
3.7.1 Calculation of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7.2 The D-K Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.7.3 Fitting D Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.7.4 Constructing Rational Perturbations . . . . . . . . . . . . . . . . . 120
3.7.5 Block Structured Norm Calculations . . . . . . . . . . . . . . . . . 121
3.8 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.8.1 Truncation and Residualization . . . . . . . . . . . . . . . . . . . . 122
3.8.2 Balanced Realizations . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.8.3 Hankel Singular Value Approximation . . . . . . . . . . . . . . . . 125
4
Demonstration Examples
127
4.1 The Himat Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.2 State-space Model of Himat . . . . . . . . . . . . . . . . . . . . . . 128
4.1.3 Creating a Weighted Interconnection Structure for Design . . . . . 131
4.1.4
H
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
∞
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4.1.5 µ Analysis of the H Controller . . . . . . . . . . . . . . . . . . . 138
∞
4.1.6 Fitting D-scales for the D-K Iteration . . . . . . . . . . . . . . . . 140
4.1.7 Design Iteration #2 . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.1.8 Simulation Comparison with a Loopshaping Controller . . . . . . . 146
4.2 A Simple Flexible Structure Example . . . . . . . . . . . . . . . . . . . . 153
4.2.1 The Control Design Problem . . . . . . . . . . . . . . . . . . . . . 153
4.2.2 Creating the Weighted Design Interconnection Structure . . . . . . 155
4.2.3 Design of an H∞ Controller . . . . . . . . . . . . . . . . . . . . . . 162
4.2.4 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2.5 D-K Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.2.6 A Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5 Bibliography
192
201
6
Function Reference
6.1 Xµ Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Xµ Subroutines and Utilities . . . . . . . . . . . . . . . . . . . . . . . . . 377
Appendices
391
A
Translation Between Matlab µ-Tools and Xµ . . . . . . . . . . . . . . . 391
A.1
A.2
Data Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
Matrix Information, Display and Plotting . . . . . . . . . . . . . . 397
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A.3
A.4
A.5
A.6
A.7
System Response Functions . . . . . . . . . . . . . . . . . . . . . . 398
System Interconnection . . . . . . . . . . . . . . . . . . . . . . . . 399
Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
H2 and H Analysis and Synthesis . . . . . . . . . . . . . . . . . 399
∞
Structured Singular Value (µ) Analysis and Synthesis . . . . . . . 400
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Chapter 1
Introduction
Xµ is a suite of Xmath functions for the modeling, analysis and synthesis of linear
robust control systems. Robust control theory has developed rapidly during the last
decade to the point where a useful set of computational tools can be used to solve a wide
range of control problems. This theory has already been applied to a wide range of
practical problems.
This manual describes the Xµ functions and presents a demonstration of their
application. The underlying theory is outlined here and further theoretical details can
be found in the many references provided.
It is assumed that the reader is familiar with the use of Xmath; the Xmath Basics
manual and the on-line demos are a good way of getting started with Xmath. A good
knowledge of control theory and application is also assumed. The more that is known
about robust control theory the better as the details are not all covered here.
1.1 Notation
Several font types or capitalization styles are used to distinguish between data objects.
The following table lists the various meanings.
1
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Notation
pdm
Meaning
Xmath parameter dependent matrix data object
Dynamic System Xmath dynamic system data object
Code examples and function names are set in typewriter font to distinguish them from
narrative text.
1.2 Manual Outline
Chapter 2 outlines the applicable robust control theory. Perturbation models and linear
fractional transformations form the basis of the modeling framework. The discussion is
aimed at an introductory level and not all of the subtleties are covered. The theory
continues with an overview of the H design technique. Again the reader is referred
∞
elsewhere for detail of the theory. The robust control methodology covered here is based
on the analysis of systems with perturbations. This is covered in some detail as such an
understanding is required for effective use of this software. Repeated analysis can be
used to improve upon the synthesis; this takes us from the standard H design method
∞
to the more sophisticated µ-synthesis techniques.
The translation between the theoretical concepts and the use of the software is made in
Chapter 3. The means of performing typical robust control calculations are discussed in
some detail. This chapter also serves to introduce the Xµ functions. The discussion is
extended to include some of the relevant Xmath functions. A prior reading of Chapter 2
is helpful for putting this material in context.
The best means of getting an idea of the use of the software is to study completed design
examples, given in Chapter 4. These currently includes a design study for an aerospace
application. Typical modeling, analysis, synthesis, and simulation studies are illustrated.
These studies can be used as initial templates for the user’s application.
Chapter 6 is a function reference guide containing a formal description of each function.
This is similar to that given via the on-line help capability. Functions are listed in
relevant groupings at the start of the chapter. This gives an overview of some of the
software capabilities.
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1.3 How to avoid really reading this Manual
The layout of the manual proceeds from introduction to background to syntax detail to
application descriptions. This may be tediously theoretical for some. If you are one of
those that considers reading the manual as the option of last resort1 then go directly to
the applications (Chapter 4). If you have no prior Xmath experience then skimming
through Chapter 3 is essential. After running the demos and getting a feel for what the
software can do look briefly through the theory section.
1And it seems that you are now exercising that option
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Chapter 2
Overview of the Underlying
Theory
2.1 Introduction
The material covered here is taken from a variety of sources. The basic approach is
described by Doyle [1, 2], and further elaborated upon by Packard [3]. Summaries have
also appeared in work by Smith [4] and others.
Motivating background can be found in the early paper by Doyle and Stein [5]. An
overview of the robust control approach, particularly for process control systems, is
given by Morari and Zafiriou [6]. The reader can also find a description of the H∞/µ
synthesis robust control approach in [7].
There are a number of descriptions of this approach to practical problems. In the last
few years a significant number of these have been described in the proceedings of the
American Control Conference (ACC) and the IEEE Control and Decision Conference
(CDC). Only some of the early illustrative examples are cited here.
Application of µ synthesis to a shuttle control subsystem is given by Doyle et al. [8].
Examples of flexible structure control are described by Balas and
coworkers [9, 10, 11, 12] and Smith, Fanson and Chu [13, 14]. There have also been
5
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several studies involving process control applications, particularly high purity distillation
columns. These are detailed by Skogestad and Morari in [15, 16, 17, 18]
Section 2.2 introduces robust control perturbation models and linear fractional
transformations. Weighted H∞ design is covered in Section 2.3. The analysis of closed
loop systems with the structured singular value (µ) is overviewed in Section 2.4.
Section 2.5 discusses µ synthesis and the D-K iteration. Model reduction is often used
to reduce the controller order prior to implementation and this is covered in Section 2.6.
2.1.1 Notation
We will use some fairly standard notation and this is given here for reference.
R
C
set of real numbers
set of complex numbers
n
R
set of real valued vectors of dimension n × 1
set of complex valued vectors of dimension n × 1
set of real valued matrices of dimension n × m
set of complex valued matrices of dimension n × m
identity matrix of dimension n × n
n
C
n×m
n×m
R
C
In
0
matrix (or vector or scalar) of zeros of appropriate dimension
n×m
The following apply to a matrix, M ∈ C
.
MT
M∗
transpose of M
complex conjugate transpose of M
|M|
absolute value of each element of M (also applies if M is a vector or scalar)
real part of M
imaginary part of M
dimensions of M
Re{M}
Im{M}
dim(M)
σmax(M) maximum singular value of M
σmin(M) minimum singular value of M
Mij
element of M in row i, column j. (also used for the i,j partition of a previously defined
partition of M)
an eigenvalue of M
λi(M)
ρ(M)
kMk
spectral radius (maxi |λi(M)|)
norm of M (see section 2.1.2 for more details)
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Figure 2.1: The generic robust control model structure
P
n
Trace(M) trace of M ( i=1 Mii)
Block diagrams will be used to represent interconnections of systems. Consider the
example feedback interconnection shown in Fig. 2.1. Notice that P has been partitioned
into four parts. This diagram represents the equations,
z
y
v
=
=
=
P11v + P12u
P21v + P22u
∆z.
This type of diagram (and the associated equations) will be used whenever the objects
P, z, y, etc., are well defined and compatible. For example P could be a matrix and z,
y, etc., would be vectors. If P represented a dynamic system then z, y, etc., would be
signals and
y = P21v + P22u,
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is interpreted to mean that the signal y is the sum of the response of system P21 to
input signal v and system P22 to input signal u. In general, we will not be specific about
the representation of the system P. If we do need to be more specific about P, then
P(s) is the Laplace representation and p(t) is the impulse response.
Note that Figure 2.1 is drawn from right to left. We use this form of diagram because it
more closely represents the order in which the systems are written in the corresponding
mathematical equations. We will later see that the particular block diagram shown in
Figure 2.1 is used as a generic description of a robust control system.
In the case where we are considering a state-space representation, the following notation
is also used. Given P(s), with state-space representation,
sx(s)
=
=
Ax(s) + Bu(s)
y(s)
Cx(s) + Du(s),
we associate this description with the notation,
ꢀ
ꢁ
A B
C D
P(s) =
.
The motivation for this notation comes from the example presented in Section 2.2.4. We
will also use this notation to for state-space representation of discrete time systems
(where s in the above is replaced by z). The usage will be clear from the context of the
discussion.
2.1.2 An Introduction to Norms
A norm is simply a measure of the size of a vector, matrix, signal, or system. We will
define and concentrate on particular norms for each of these entities. This gives us a
formal way of assessing whether or not the size of a signal is large or small enough. It
allows us to quantify the performance of a system in terms of the size of the input and
output signals.
Unless stated otherwise, when talking of the size of a vector, we will be using the
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Euclidean norm. Given,
x1
.
.
.
x =
,
xn
the Euclidean (or 2-norm) of x, denoted by kxk, is defined by,
!
1/2
n
X
kxk =
|xi|
.
i=1
Many other norms are also options; more detail on the easily calculated norms can be
found in the on-line help for the norm function. The term spatial-norm is often applied
when we are looking at norms over the components of a vector.
Now consider a vector valued signal,
x1(t)
.
.
.
x(t) =
.
xn(t)
As well as the issue of the spatial norm, we now have the issue of a time norm. In the
theory given here, we concentrate on the 2-norm in the time domain. In otherwords,
ꢂ
Z
ꢃ
1/2
∞
2
kxi(t)k =
|xi(t)| dt
.
−∞
This is simply the energy of the signal. This norm is sometimes denoted by a subscript
of two, i.e. kxi(t)k2. Parseval’s relationship means that we can also express this norm in
the Laplace domain as follows,
ꢂ
ꢃ
Z
1/2
∞
1
2π
2
kxi(s)k =
|xi(ω)| dω
.
−∞
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For persistent signals, where the above norm is unbounded, we can define a power norm,
!
1/2
Z
T
1
2T
2
kxi(t)k = lim
|xi(t)| dt
.
(2.1)
T →∞
−T
The above norms have been defined in terms of a single component, xi(t), of a vector
valued signal, x(t). The choice of spatial norm determines how we combine these
components to calculate kx(t)k. We can mix and match the spatial and time parts of the
norm of a signal. In practice it usually turns out that the choice of the time norm is
more important in terms of system analysis. Unless stated otherwise, kx(t)k implies the
Euclidean norm spatially and the 2-norm in the time direction.
Certain signal spaces can be defined in terms of their norms. For example, the set of
signals x(t), with kx(t)k2 finite is denoted by L2. The formal definition is,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
L2 = x(t) kx(t)k < ∞
.
ꢅ
A similar approach can be taken in the discrete-time domain. Consider a sequence,
{x(k)}∞k=0, with 2-norm given by,
!
1/2
∞
X
2
kx(k)k2 =
|x(k)|
.
k=0
A lower case notation is used to indicate the discrete-time domain. All signals with finite
2-norm are therefore,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
l2 = x(k), k = 0, . . . , ∞ kx(k)k2 < ∞
.
ꢅ
We can essentially split the space L2 into two pieces, H2 and H2⊥. H2 is the set of
elements of L2 which are analytic in the right-half plane. This can be thought of as
those which have their poles strictly in the left half plane; i.e. all stable signals.
Similarly, H2⊥ are all signal with their poles in the left half plane; all strictly unstable
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signals. Strictly speaking, signals in H2 or H⊥ are not defined on the ω axis. However
we usually consider them to be by taking a li2mit as we approach the axis.
A slightly more specialized set is RL2, the set of real rational functions in L2. These are
strictly proper functions with no poles on the imaginary axis. Similarly we can consider
RH2 as strictly proper stable functions and RH⊥2 as strictly proper functions with no
poles in Re(s)< 0. The distinction between RL2 and L2 is of little consequence for the
sorts of analysis we will do here.
The concept of a unit ball will also come up in the following sections. This is simply the
set of all signals (or vectors, matrices or systems) with norm less than or equal to one.
The unit ball of L2, denoted by BL2, is therefore defined as,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
BL2 = x(t) kx(t)k2 < 1
.
ꢅ
Now let’s move onto norms of matrices and systems. As expected the norm of a matrix
gives a measure of its size. We will again emphasize only the norms which we will
consider in the following sections. Consider defining a norm in terms of the maximum
gain of a matrix or system. This is what is known as an induced norm. Consider a
matrix, M, and vectors, u and y, where
y = Mu.
Define, kMk, by
kyk
kMk = max
.
u,kuk<∞ kuk
Because M is obviously linear this is equivalent to,
kMk = max kyk.
u,kuk=1
The properties of kMk will depend on how we define the norms for the vectors u and y.
If we choose our usual default of the Euclidean norm then kMk is given by,
kMk = σmax(M),
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where σmax denotes the maximum singular value. Not all matrix norms are induced
from vector norms. The Froebenius norm (square root of the sum of the squares of all
matrix elements) is one such example.
Now consider the case where P(s) is a dynamic system and we define an induced norm
from L2 to L2 as follows. In this case, y(s) is the output of P(s)u(s) and
ky(s)k2
kP(s)k = max
.
u(s)∈L2 ku(s)k2
Again, for a linear system, this is equivalent to,
kP(s)k = max ky(s)k2 .
u(s)∈BL2
This norm is called the ∞-norm, usually denoted by kP(s)k∞. In the single-input,
single-output case, this is equivalent to,
kP(s)k∞ = ess sup |P(ω)|.
ω
This formal definition uses the term ess sup, meaning essential supremum. The
“essential” part means that we drop all isolated points from consideration. We will
always be considering continuous systems so this technical point makes no difference to
us here. The “supremum” is conceptually the same as a maximum. The difference is
that the supremum also includes the case where we need to use a limiting series to
approach the value of interest. The same is true of the terms “infimum” (abbreviated to
“inf”) and “minimum.” For practical purposes, the reader can think instead in terms of
maximum and minimum.
Actually we could restrict u(s) ∈ H2 in the above and the answer would be the same. In
other words, we can look over all stable input signals u(s) and measure the 2-norm of
the output signal, y(s). The subscript, ∞, comes from the fact that we are looking for
the supremum of the function on the ω axis. Mathematicians sometimes refer to this
norm as the “induced 2-norm.” Beware of the possible confusion when reading some of
the mathematical literature on this topic.
If we were using the power norm above (Equation 2.1) for the input and output norms,
the induced norm is still kP(s)k∞.
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The set of all systems with bounded ∞-norm is denoted by L∞. We can again split this
into stable and unstable parts. H∞ denotes the stable part; those systems with |P(s)|
finite for all Re(s)> 0. This is where the name “H∞ control theory” originates, and we
often call this norm the H∞-norm. Again we can restrict ourselves to real rational
functions, so RL∞ is the set of proper transfer functions with no poles on the ω axis.
Similary, RH∞ is the set of proper, stable transfer functions.
Again, we are free to choose a spatial norm for the input and output signals u(s) and
y(s). In keeping with our above choices we will choose the Euclidean norm. So if P(s) is
a MIMO system, then,
kP(s)k∞ = sup σmax[P(ω)].
ω
There is another choice of system norm that will arise in the following sections. This is
the H2-norm for systems, defined as,
ꢂ
ꢃ
Z
1/2
∞
1
2π
kP(s)k2 =
−∞ Trace[P(ω)∗P(ω)]dω
,
where P(ω)∗ denotes the conjugate transpose of P(ω) and the trace of a matrix is the
sum of its diagonal elements. This norm will come up when we are considering linear
quadratic Gaussian (LQG) problems.
2.2 Modeling Uncertain Systems
2.2.1 Perturbation Models for Robust Control
A simple example will be used to illustrate the idea of a perturbation model. We are
interested in describing a system by a set of models, rather than just a nominal model.
Our uncertainty about the physical system will be represented in an unknown
component of the model. This unknown component is a perturbation, ∆, about which
we make as few assumptions as possible; maximum size, linearity, time-invariance, etc..
Every different perturbation, ∆, gives a slightly different system model. The complete
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robust control model is therefore a set description and we hope that some members of
this set capture some of the uncertain or unmodeled aspects of our physical system.
For example, consider the “uncertain” model illustrated in Figure 2.2. This picture is
equivalent to the input-output relationship,
y = [ (I + ∆Wm)Pnom ] u.
(2.2)
Figure 2.2: Generic output multiplicative perturbation model
In this figure, ∆, Wm and Pnom are dynamic systems. The most general form for the
theory can be stated with these blocks as elements of H∞. For the purposes of
calculation we will be dealing with Xmath Dynamic Systems, and in keeping with this
we will tend to restrict the theoretical discussion to RH∞, stable, proper real rational
transfer function matrices.
The only thing that we know about the perturbation, ∆, is that k∆k∞ ≤ 1. Each ∆,
with k∆k∞ ≤ 1 gives a different transfer function between u and y. The set of all
possible transfer functions, generated in this manner, is called P. More formally,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
P = (I + ∆Wm)Pnom k∆k∞ ≤ 1
.
(2.3)
ꢅ
Now we are looking at a set of possible transfer functions,
y(s) = P(s)u(s),
where P(s) ∈ P.
Equation 2.2 represents what is known as a multiplicative output perturbation
structure. This is perhaps one of the easiest to look at initially as W(s) can be viewed
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as specifying a maximum percentage error between Pnom and every other element of P.
The system Pnom(s) is the element of P that comes from ∆ = 0 and is called the
nominal system. In otherwords, for ∆ = 0, the input-output relationship is
y(s) = Pnom(s) u(s). As ∆ deviates from zero (but remains bounded in size), the
nominal system is multiplied by (I + ∆Wm(s)). Wm(s) is a frequency weighting function
which allows us the specify the maximum effect of the perturbation for each frequency.
Including Wm(s) allows us to model P with ∆ being bounded by one. Any
normalization of ∆ is simply included in Wm(s).
We often assume that ∆ is also linear and time-invariant. This means that ∆(ω) is
simply an unknown, complex valued matrix at each frequency, ω. If k∆k∞ ≤ 1, then, at
each frequency, σmax(∆(ω)) ≤ 1. Section 2.2.3 gives a further discussion on the pros
and cons of considering ∆ to be linear, time-invariant.
Now consider an example of this approach from a Nyquist point of view. A simple first
order SISO system with multiplicative output uncertainty is modeled as
ꢀ
ꢁ
y(s) = (I + Wm(s)∆)Pnom(s) u(s),
where
1 + 0.05s
1 + s
0.1 + 0.2s
1 + 0.05s
Pnom(s) =
and Wm(s) =
.
Figure 2.3 illustrates the set of systems generated by a linear time-invariant ∆,
k∆k∞ ≤ 1.
At each frequency, ω, the transfer function of every element of P, lies within a circle,
centered at Pnom(ω), of radius |Pnom(ω)Wm(ω)|. Note that for certain frequencies the
disks enclose the origin. This allows us to consider perturbed systems that are
non-minimum phase even though the nominal system is not.
It is worth pointing out that P is still a model; in this case a set of regions in the
Nyquist plane. This is model set is now able to describe a larger set of system behaviors
than a single nominal model. There is still an inevitable mismatch between any model
(robust control model set or otherwise) and the behaviors of a physical system.
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1
0.5
0
-0.5
-1
-0.5
0
0.5
1
1.5
Real
Figure 2.3: Nyquist diagram of the set of systems, P
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Figure 2.4: Unity gain negative feedback for the example system, P0 + ∆Wa
2.2.2 Linear Fractional Transformations
A model is considered to be an interconnection of lumped components and perturbation
blocks. In this discussion we will denote the input to the model by u, which can be a
vector valued signal representing input signals such as control inputs, disturbances, and
noise. The outputs signal, denoted in this discussion by y, are also vector valued and can
represent system outputs and other variables of interest.
In order to treat large systems of interconnected components, it is necessary to use a
model formulation that is general enough to handle interconnections of systems. To
illustrate this point consider an affine model description:
y = (P0 + ∆Wa)u, k∆k∞ ≤ 1,
(2.4)
where u is the input and y is the output. ∆ again represents an unknown but bounded
perturbation. This form of perturbed model is known as an additive perturbation
description. While such a description could be applied to a large class of linear systems,
it is not general enough to describe the interconnection of models. More specifically, an
interconnection of affine models is not necessarily affine. To see this, consider unity gain
positive feedback around the above system. This is illustrated in Figure 2.4.
The new input-output transfer function is
y = (P0 + ∆Wa)[I + (P0 + ∆Wa)]−1 r.
(2.5)
It is not possible to represent the new system with an affine model. Note that stability
questions arise from the consideration of the invertibility of [I + (P0 + ∆Wa)].
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Figure 2.5: Generic LFT model structure including perturbations,∆
A generic model structure, referred to as a linear fractional transformation (LFT),
overcomes the difficulties outlined above. The LFT model is equivalent to the
relationship,
ꢇ
ꢈ
y = P21∆(I − P11∆)−1P12 + P22 u,
(2.6)
where the ∆ is the norm bounded perturbation. Figure 2.5 shows a block diagram
equivalent to the system described by Equation 2.6. Because this form of interconnection
is widely used, we will give it a specific notation. Equation 2.6 is abbreviated to,
y = Fu(P, ∆)u.
The subscript, u, indicates that the ∆ is closed in the upper loop. We will also use
Fl(., .) when the lower loop is closed.
In this figure, the signals, u, y, z and v can all be vector valued, meaning that the
partitioned parts of P, (P11, etc.) can themselves be matrices of transfer functions.
To make this clear we will look at the perturbed system example, given in Equation 2.4,
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in an LFT format. The open-loop system is described by,
y = Fu(Polp, ∆)u,
where
ꢀ
ꢁ
0 Wa
I P0
Polp
=
.
The unity gain, negative feedback configuration, illustrated in Figure 2.4 (and given in
Equation 2.5) can be described by,
y = Fu(Gclp, ∆)r,
where
ꢀ
ꢁ
−Wa(I + P0)−1 Wa(I + P0)−1
Gclp
=
(I + P0)−1
P0(I + P0)−1
Figure 2.5 also shows the perturbation, ∆ as block structured. In otherwords,
∆ = diag(∆1, . . . , ∆m).
(2.7)
This allows us to consider different perturbation blocks in a complex interconnected
system. If we interconnect two systems, each with a ∆ perturbation, then the result can
always be expressed as an LFT with a single, structured perturbation. This is a very
general formulation as we can always rearrange the inputs and outputs of P to make ∆
block diagonal.
The distinction between perturbations and noise in the model can be seen from both
Equation 2.6 and Figure 2.5. Additive noise will enter the model as a component of u.
The ∆ block represents the unknown but bounded perturbations. It is possible that for
some ∆, (I − P11∆) is not invertible. This type of model can describe nominally stable
systems which can be destabilized by perturbations. Attributing unmodeled effects
purely to additive noise will not have this characteristic.
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The issue of the invertibility of (I − P11∆) is fundamental to the study of the stability of
a system under perturbations. We will return to this question in much more detail in
Section 2.4. It forms the basis of the µ analysis approach.
Note that Equation 2.7 indicates that we have m blocks, ∆i, in our model. For
notational purposes we will assume that each of these blocks is square. This is actually
without loss of generality as in all of the analysis we will do here we can square up P by
adding rows or columns of zeros. This squaring up will not affect any of the analysis
results. The software actually deals with the non-square ∆ case; we must
specify the input and output dimensions of each block.
The block structure is a m-tuple of integers, (k1, . . . , km), giving the dimensions of each
∆i block. It is convenient to define a set, denoted here by ∆, with the appropriate block
structure representing all possible ∆ blocks, consistent with that described above. By
this it is meant that each member of the set of ∆ be of the appropriate type (complex
matrices, real matrices, or operators, for example) and have the appropriate dimensions.
In Figure 2.5 the elements P11 and P12 are not shown partitioned with respect to the
∆i. For consistency the sum of the column dimensions of the ∆i must equal the row
dimension of P11. Now define ∆ as
ꢅ
ꢅ
n
o
∆ = diag (∆ , . . . , ∆ ) dim(∆ ) = k × k
.
ꢅ
1
m
i
i
i
It is assumed that each ∆i is norm bounded. Scaling P allows the assumption that the
norm bound is one. If the input to ∆i is zi and the output is vi, then
kvik = k∆izik ≤ kzik.
It will be convenient to denote the unit ball of ∆, the subset of ∆ norm bounded by
one, by B∆. More formally
ꢅ
ꢅ
n
o
B∆ = ∆ ∈ ∆ k∆k ≤ 1 .
ꢅ
Putting all of this together gives the following abbreviated representation of the
perturbed model,
y = Fu(P, ∆)u,
∆ ∈ B∆.
(2.8)
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Figure 2.6: Example model: multiplicative output perturbation with weighted output
noise
References to a robust control model will imply a description of the form given in
Equation 2.8.
As a example, consider one of the most common perturbation model descriptions,
illustrated in Figure 2.6. This model represents a perturbed system with bounded noise
at the output.
The example model is given by,
y = Wnw + (I + ∆Wu)Pnomu.
The system Wn is a frequency dependent weight on the noise signal, w. This allows us
to use a normalized representation for w. In other words the model includes the
assumption that kwk∞ ≤ 1. Similarly, we assume that k∆k∞ ≤ 1 and Wu is a frequency
dependent weight which specifies the contribution of the perturbation at each frequency.
In a typical model Wn will be small (assuming that the noise is small compared to the
nominal output) and Wu will increase at high frequencies (to capture the likely case that
we know less about the model at higher frequencies). The LFT representation of this
model is,
h
i
w
u
y = Fu(P, ∆)
,
where
"
#
0
0
Wn
WuPnom
Pnom
P =
I
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Robust control models are therefore set descriptions. In the analysis of such models it is
also assumed that the unknown inputs belong to some bounded set. Several choices of
set for the unknown signals can be made, leading to different mathematical problems for
the analysis. Unfortunately not all of them are tractable. The following section discusses
the assumptions typically applied to the robust control models.
2.2.3 Assumptions on P, ∆, and the unknown signals
It will be assumed that the elements of P are either real-rational transfer function
matrices or complex valued matrices. The second case arises in the frequency by
frequency analysis of systems.
In modeling a system, P22 defines the nominal model. Input/output effects not
described by the nominal model can be attributed to either unknown signals which are
components of the model input (w in the previous example), or the perturbation ∆.
Unmodeled effects which can destabilize a system should be accounted for in ∆. The ∆
can loosely be considered as accounting for the following. This list is by no means
definitive and is only included to illustrate some of the physical effects better suited to
description with ∆.
• Unmodeled dynamics. Certain dynamics may be difficult to identify and there
comes a point when further identification does not yield significant design
performance improvement.
• Known dynamics which have been bounded and included in ∆ to simplify the
model. As the controller complexity depends on the order of the nominal model a
designer may not wish to explicitly include all of the known dynamics.
• Parameter variations in a differential equation model. For example linearization
constants which can vary over operating ranges.
• Nonlinear or inconsistent effects. At some point a linear model will no longer
account for the residual differences between the behaviors of the model and the
physical system.
Several assumptions on ∆ are possible. In the most general case ∆ is a bounded
operator. Alternatively ∆ can be considered as a linear time varying multiplier. This
assumption can be used to capture nonlinear effects which shift energy between
frequencies. Analysis and synthesis are possible with this assumption; Doyle and
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Packard [19] discuss the implications of this assumption on robust control theory and we
briefly touch upon this in Section 2.4.6. The most common assumption is that ∆ is an
unknown, norm-bounded, linear time-invariant system.
Systems often do not fall neatly into one of the usual choices of ∆ discussed above.
Consider a nonlinear system linearized about an operating point. If a range of operation
is desired then the linearization constants can be considered to lie within an interval.
The model will have a ∆ block representing the variation in the linearization constants.
If this is considered to be a fixed function of frequency then the model can be considered
to be applicable for small changes about any operating point in the range. The precise
meaning of small will depend on the effect of the other ∆ blocks in the problem.
If the ∆ block is assumed to be time-varying then arbitrary variation is allowed in the
operating point. However this variation is now arbitrarily fast, and the model set now
contains elements which will not realistically correspond to any observed behavior in the
physical system.
The robust control synthesis theory gives controllers designed to minimize the maximum
error over all possible elements in the model set. Including non-physically motivated
signals or conditions can lead to a conservative design as it may be these signals or
conditions that determine the worst case error and consequently the controller.
Therefore the designer wants a model which describes all physical behaviors of the
system but does not include any extraneous elements.
The designer must select the assumptions on P and ∆. An inevitable tradeoff arises
between the ideal assumptions given the physical considerations of the system, and those
for which good synthesis techniques exist.
The most commonly used assumption is that ∆ is a linear time invariant system. This
allows us to consider the interconnection, Fu(P, ∆), from a frequency domain point of
view. At each frequency ∆ can be taken as an unknown complex valued matrix of norm
less than or equal to one. This leads to analyses (covered in Section 2.4) involving the
complex structured singular value. The following section discusses more complicated
block structures and their use in modeling uncertain systems.
2.2.4 Additional Perturbation Structures
Equation 2.7 introduced a perturbation structure, ∆ containing m perturbation blocks,
∆i. This form of perturbation is applicable to a wide range of models for uncertain
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systems. We will now look at other possible perturbation structures. For more detail on
these structures (in the complex case) refer to Packard and Doyle [20].
Consider a blocks which are of the form scalar × identity, where the scalar is unknown.
In the following we will include q of these blocks in ∆. The definition of ∆ is therefore
modified to be,
ꢅ
ꢅ
n
o
∆ = diag(δ I , . . . , δ I , ∆ , . . . , ∆ ) dim(I ) = l × l , dim(∆ ) = k × k .(2.9)
ꢅ
1
1
q
q
1
m
j
j
j
i
i
i
The block structure now contains the dimension of the q scalar × identity blocks and the
m full blocks. The block structure is therefore, (l1, . . . , lq, k1, . . . , km). If
dim(∆) = n × n, then these dimensions must be consistent. In otherwords,
q
m
X
X
lj +
ki = n.
j=1
i=1
Note that this block structure collapses to the previously defined structure
(Equation 2.7) when q = 0.
The most obvious application of a repeated scalar block structure occurs when we know
that perturbations occurring in several places in a system are identical (or perhaps just
correlated). For example, dynamic models of aircraft often have the altitude (or
dynamic pressure) occuring in several places in the model. Naturally the same value
should be used in each place and if we model the altitude as an LFT parameter then the
repeated scalar × identity approach is the most appropriate.
This structure also allows us to express uncertain state-space models as LFTs. To
illustrate this consider the following discrete time system.
x(k + 1)
=
=
Ax(k) + Bu(k)
y(k)
Cx(k) + Du(k).
This digital system has transfer function,
P(z)
=
C(zI − A)−1B + D
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=
=
Cz−1(I − z−1A)−1B + D
Fu(Pss, z−1I),
where Pss is the real valued matrix,
ꢀ
ꢁ
A B
C D
Pss
=
,
and the scalar × identity, z−1I, has dimension equal to the state dimension of P(z).
This is now in the form of an LFT model with a single scalar × identity element in the
upper loop.
One possible use of this is suggested by the following. Define,
ꢅ
ꢅ
ꢉ
∆ = δInx δ ∈ C ,
where nx is the state dimension. The set of models,
Fu(Pss, ∆), ∆ ∈ B∆,
is equivalent to P(z), |z| ≥ 1. This hints at using this formulation for a stability analysis
of P(z). This is investigated further in Section 2.4.6.
In the analyses discussed in Section 2.4 we will concentrate on the assumption that ∆ is
complex valued at each frequency. For some models we may wish to restrict ∆ further.
The most obvious restriction is that some (or all) of the ∆ blocks are real valued. This is
applicable to the modeling of systems with uncertain, real-valued, parameters. Such
models can arise from mathematical system models with unknown parameters.
Consider, for example a very simplified model of the combustion characteristics of an
automotive engine. This is a simplified version of the model given by Hamburg and
Shulman [21]. The system input to be considered is the air/fuel ratio at the carburettor.
The output is equivalent to the air/fuel ratio after combustion. This is measured by an
oxygen sensor in the exhaust. Naturally, this model is a strong function of the engine
speed, v (rpm). We model the relationship as,
ꢂ
ꢃ
0.9
0.1
y = e−T s
+
u,
d
1 + Tcs 1 + s
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where the transport delay, Td, and the combustion lag, Tc, are approximately,
252
v
202
v
Td =
and Tc =
.
For the purposes of our example we want to design an air/fuel ratio controller that
works for all engine speeds in the range 2,000 to 6,000 rpm. We will use a first order
Pad´e approximation for the delay and express the Td and Tc relationships in an LFT
form with a normalized speed deviation, δv.
The dominant combustion lag can be expressed as an LFT on 1/Tc as follows,
0.9
1 + Tcs
=
Fu(Ptc, Tc−1),
where
−1
1
s
Ptc
=
.
0.9
0
s
Note that Tc−1 is easily modeled in terms of δv,
1
Tc
4000 + 2000δv
=
,
δv ∈ R, |δv| ≤ 1.
202
The Pad´e approximation (with delay time, Td) is given by
ꢊ
ꢋ
e−T s ≈ Fu Pdelay, Td−1
,
d
where,
−2
1
s
Pdelay
=
.
4
−1
s
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Putting all the pieces together gives an engine model in the following fractional form.
P(s) = Fu(Pmod, ∆),
where
7.14(s − 19.8)
141.5(1 + 1.006s)
−15.87
s2
s
s(s + 1)
−9.9
Pmod
=
,
0
9.9
s
−0.9(s − 15.8)(s − 19.8) −17.82(s − 15.8)(1 + 1.006s)
s(s + 1)
−27.75
s3
s
and ∆ ∈ B∆, with the structure defined as,
ꢅ
ꢅ
ꢉ
∆ = δvI2 δv ∈ R
.
To capture the effects of unmodeled high frequency dynamics we will also include an
output multiplicative perturbation. If the output multiplicative weight is Wm(s) then
the complete open-loop model is,
P(s) = Fu(Pmod, ∆),
where,
(2.10)
7.14(s − 19.8)
141.5(1 + 1.006s)
−15.87
0
s2
s
s(s + 1)
−9.9
0
0
0
9.9
s
Pmod
=
−0.9(s − 15.8)(s − 19.8)
−17.82(s − 15.8)(1 + 1.006s)
−27.75
s3
s
s(s + 1)
−0.9(s − 15.8)(s − 19.8)
−17.82(s − 15.8)(1 + 1.006s)
−27.75
Wm(s)
s3
s
s(s + 1)
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and ∆ ∈ B∆, with the structure defined as,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
∆ = diag(δvI2, ∆1) δv ∈ R, ∆1 ∈ C
.
ꢅ
Note that this is an LFT with a repeated real-valued parameter, δv (|δv| ≤ 1), and a
complex perturbation, ∆1 (|∆1| ≤ 1).
Note that as R ⊂ C, assuming that ∆ ∈ Cn×n, always covers the case where some of the
∆i (or δj) are more appropriately modeled as real-valued. However this may be
potentially conservative as the ∆ ∈ Cn×n, allows many more systems in the model set.
In this case it would be somewhat better to consider combining the effects of δv and ∆1
into a single complex valued ∆ with an appropriate weight.
In principle, if we have additional information about the system (some δj ∈ R, for
example) then we should use this information. Performing analyses with real valued
perturbations is currently at the forefront of the structured singular value research. We
will return to this issue in more detail when we cover the analysis methods (Section 2.4).
2.2.5 Obtaining Robust Control Models for Physical Systems
Obtaining a model of the above form is where the real engineering comes in. A designer
must model and identify the physical system to arrive at such a model. This is usually
an iterative process whereby designs are performed and then tested on the system. In
this way the designer often obtains a feeling for the adequacy of the model.
The best way of studying the modeling problem is to look at the documented
experiences of others applying these approaches; particularly in similar applications. The
citations given in the Section 2.1 will be useful in this regard. There are also approaches
addressing the problem of obtaining LFT models from descriptions with variable
state-space matrix coefficients [22, 23]. In the area of SISO process control, Laughlin et
al. [24] describe the relationship between uncertainties in time constant, delay, and gain,
and a suitable ∆ weighting. Models of this form are often applicable to process control.
There is little formal theory addressing the robust control modeling problem although
this is an area of increasing interest. A recent workshop proceedings volume on the
subject is an excellent reference for those interested in this area [25]. Other references
can be found in the review article by Gevers [26].
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H∞
H2
An area of work, known as identification in H∞, looks at experimental identification
techniques which minimize the worst case H∞ error between the physical system and
the model. The following works address this issue: [27, 28, 29, 30, 31, 32, 33, 34, 35].
Applying the more standard, probabilistically based, identification techniques to
uncertain systems is also receiving attention. Relevant work in this area is described
in: [36, 37, 38, 39]
Model validation is the experimental testing of a given robust control model. This can
be useful is assessing model quality. This work is covered in the
following: [4, 40, 41, 42, 43, 44, 45, 46]. An experimental example is described by
Smith [47].
The problems of identifying model parameters in an uncertain model is discussed further
in [48, 49, 50]. A nonlinear ad-hoc approach for obtaining suitable multiplicative
perturbation models for certain classes of systems is given in [51].
Several researchers are also formalizing the interplay between identification and design
in iterative approaches. In practical situations the designer usually ends up with ad-hoc
identification/design iterations. The work in this area is described in [52, 53, 54, 55, 56].
On reading the above works, one will get the impression that this area is the most
poorly developed of the current robust control theory. In obtaining these models
engineering judgement is of paramount importance. The users of this software are
encouraged to document their experiences and bring this work to the authors’ attention.
2.3 H∞ and H2 Design Methodologies
The generic synthesis configuration is illustrated in LFT form in Figure 2.7. Here P(s)
is referred to as the interconnection structure. The objective is to design K(s) such that
the closed loop interconnection is stable and the resulting transfer function from w to e
(denoted by G(s)),
e
=
=
Fl[P(s), K(s)]w,
G(s)w,
satisfies a norm objective.
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Figure 2.7: LFT configuration for controller synthesis, G(s) = Fl[P(s), K(s)]
Note that the interconnection structure, P(s), given here, differs from that discussed in
the previous section. Here we set up P(s) so that the input, w, is the unknown signals
entering our system. Typical examples would be sensor noise, plant disturbances or
tracking commands. The output, e, represent signals that we would like to make small.
In an engineering application these could include actuator signals and tracking errors.
The signal y is the measurement available to the controller, K(s). In any realistic
problem, some weighted component of w would be added to y to model sensor noise.
The output of the controller, u, is our actuation input to the system. Again, a
reasonable engineering problem would include a weighted u signal as a component of the
penalty output, e.
The interconnection structure, P(s), also contains any frequency weightings on the
signals e and w. Weightings on components of e are used to determine the relative
importance of the various error signals. Weight functions on w indicate the relative
expected size of the unknown inputs.
Xµ provides functions to calculate the controllers minimizing either the H2 or H∞ norm
of G(s). We will cover both of these approaches in the context of the design problem
illustrated in Figure 2.7.
Note that neither of these design approaches takes advantage of any information about
structured perturbations occuring within the model. The following discussion can be
considered as applying to a nominal design problem. Section 2.5 uses D-K iteration to
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H∞
H2
extend these approaches to the case where P(s) is replaced by Fu(P(s), ∆), ∆ ∈ B∆.
2.3.1 H∞ Design Overview
Again, recall from Section 2.1.2, the H∞ is norm of G(s) is,
kG(s)k∞ = sup σmax[G(ω)].
ω
The H∞ norm is the induced L2 to L2 norm. Therefore minimizing the H∞ norm of
G(s) will have the effect of minimizing the worst-case energy of e over all bounded
energy inputs at w.
Consider γ(K) to be the closed loop H∞ norm achieved for a particular controller K. In
other words,
γ(K) = kFl(P, K)k∞.
There is a choice of controller, K, which minimizes γ(K). This is often referred to as the
optimal value of γ and is denoted by γopt. Furthermore, there is no stabilizing controller
which satisfies,
kG(s)k∞ < γopt
.
In a particular design problem, γopt is not known a priori. Therefore the functions
calculating the H∞ controller use some form of optimization to obtain a value of γ close
to γopt
.
The first approaches to the solution of this problem were described by Doyle [1]. The
book by Francis [57] gives a good overview of the early version of this theory. A
significant breakthrough was achieved with the development of state-space calculation
techniques for the problem. These are discussed in the paper colloquially known as
DGKF [58]. The algorithmic details are actually given by Glover and Doyle [59].
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2.3.2 Assumptions for the H∞ Design Problem
There are several assumptions required in order to achieve a well-posed design problem.
The DGKF paper gives a state-space solution to the H∞ design problem and we will use
a similar notation here.
Consider the open loop state-space representation of P(s), partitioned according to the
signals shown in Figure 2.7,
A
C1
C2
B1 B2
D11 D12
D21 D22
P(s) =
.
(2.11)
We will assume that P(s) is a minimal representation. The following assumptions are
required for a well-posed problem.
(i) (A, B2) is stabilizable and (C2, A) is detectable;
(ii) D12 and D21 are full rank;
(iii) The matrix,
ꢀ
ꢁ
A − ωI B2
C1 D12
,
has full column rank for all ω ∈ R;
(iv) The matrix,
ꢀ
ꢁ
A − ωI B1
C2 D21
,
has full row rank for all ω ∈ R.
Item (i) is required so that input-output stability is equivalent to internal stability. If it
is not satisfied then there are unstable modes which cannot be stabilized by any K(s).
Items (ii) and (iii) mean that, at every frequency, there is no component of the output
signal, e, that cannot be influenced by the controller. Similarly, items (ii) and (iv) mean
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H∞
H2
that the effect of all disturbances, w, at every frequency, can be measured by the
controller. If either of these conditions are not met then the problem could be ill-posed.
It is possible to violate these conditions by using pure integrators as design weights.
While this could still give a meaningful design problem, solution via the state-space H∞
approach requires that an approximation be used for the integrator weight. If item (iii)
or (iv) is violated at ω = 0, then the integrator should be replaced with very low
frequency pole.
2.3.3 A Brief Review of the Algebraic Riccati Equation
Solution of the H∞ design problem requires the solution of coupled Algebraic Riccati
Equations (AREs). This is illustrated in more detail in the next section. Here we give a
very brief review of the Riccati equation and the most common solution techniques.
Some knowledge of this area is helpful because the design software displays variables
related to the Riccati solutions and the user has the option of adjusting several software
tolerances relating to these solutions. The notation used here comes from DGKF [58].
The matrix equation,
AT X + XA + XRX − Q = 0,
is an ARE. Given A, R and Q (with R and Q symmetric), we are interested in finding a
symmetric positive definite solution, X. In other words, X = XT ≥ 0. With this ARE
we associate a Hamiltonian matrix, denoted by H,
ꢀ
ꢁ
A
R
H =
.
Q −AT
If dim(A) = n × n, then dim(H) = 2n × 2n. Assume that H has no ω axis eigenvalues.
The structure of H means that it has n stable (Re{s} < 0) and n unstable (Re{s} > 0)
eigenvalues.
Now consider finding a basis for the stable eigenvalues. Stacking the basis vectors
together will give a 2n × n matrix,
h
i
X1
X2
.
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We have partitioned the matrix into two n × n blocks, X1 and X2. If X1 is invertible,
then
X = X2X1−1
,
is the unique, stabilizing solution to the ARE. The ability to form X doesn’t depend on
the particular choice of X1 and X2.
Given a Hamiltonian, H, we say that H ∈ dom(Ric) if H has no ω axis eigenvalues and
the associated X1 matrix is invertible. Therefore, if H ∈ dom(Ric), we can obtain a
unique stabilizing solution, X. This mapping, from H to X, is often written as the
function, X = Ric(H).
To give an idea of the application of the ARE consider the following lemma (taken from
DGKF).
Lemma 1 Suppose H ∈ dom(Ric) and X = Ric(H). Then:
a) X is symmetric;
b) X satisfies the ARE,
AT X + XA + XRX − Q = 0;
c) A + RX is stable.
This is of course the well know result relating AREs to the solution of stabilizing state
feedback controllers.
AREs can also be used in calculating the H∞-norm of a state-space system. The
approach outlined here is actually that used in the software for the calculation of
kP(s)k∞. Consider a stable system,
ꢀ
ꢁ
A B
C 0
P(s) =
.
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H∞
H2
Choose γ > 0 and form the following Hamiltonian matrix,
ꢀ
ꢁ
A
−CT C
γ
−2BBT
−AT
H =
.
The following lemma gives a means of checking whether or not kP(s)k∞ < γ. A proof of
this lemma is given in DGKF although it is based on the work of Anderson [60],
Willems [61] and Boyd et al. [62].
Lemma 2 The following conditions are equivalent:
a) kP(s)k∞ < γ;
b) H has no eigenvalues on the ω axis;
c) H ∈ dom(Ric);
d) H ∈ dom(Ric) and Ric(H) ≥ 0 (if (C,A) is observable then Ric(H) > 0).
As the above illustrates, AREs play a role in both stabilization and H∞-norm
calculations for state-space systems. Before giving more detail on the H∞ design
problem (Section 2.3.4), we will discuss some of the issues that arise in the practical
calculation of ARE solutions.
We can summarize an ARE solution method as follows:
(i) Form the Hamiltonian, H.
(ii) Check that H has no ω axis eigenvalues.
(iii) Find a basis for the stable subspace of H.
(iv) Check that X1 is invertible.
(v) Form X = X2X1−1
.
The first issue to note is that it is difficult to numerically determine whether or not H
has ω axis eigenvalues. A numerical calculation of the eigenvalues is unlikely to give
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any with a zero real part. In practice we must use a tolerance to determine what is
considered as a zero real part.
Finding a basis for the stable subspace of H involves either an eigenvalue or Schur
decomposition. Numerical errors will be introduced at this stage. In most cases using an
eigenvalue decomposition is faster and less accurate than using a Schur decomposition.
Similarly, forming X = X2X1−1 will also introduce numerical errors. The Schur solution
approach, developed by Laub et al. [63, 64, 65], is currently the best numerical approach
to solving the ARE and is used in the software as the default method. An overview of
invariant subspace methods for ARE solution is given by Laub [66]. Accurate solution of
the ARE is still very much an active area of research.
2.3.4 Solving the H∞ Design Problem for a Special Case
We will now look at the H∞ design problem for a simplifying set of assumptions. The
general problem (with assumptions given in Section 2.3.2) can be transformed into the
simplified one given here via scalings and other transformations. The simplified problem
illustrates the nature of the solution procedure and is actually the problem studied in
DGKF. The formulae for the general problem are given in Glover and Doyle [59]. The
software solves the general problem.
Consider the following assumptions, with reference to the system in Equation 2.11:
(i) (A,B1) stabilizable and (C1,A) detectable;
(ii) (A,B2) stabilizable and (C2,A) detectable;
(iii) D1T2[ C1 D12 ] = [ 0 I ];
ꢀ
ꢁ
h
i
B1
D21D2T1
0
(iv)
=
;
I
(v) D11 = D22 = 0.
Assumption (i) is included in DGKF for technical reasons. The formulae are still correct
if it is violated. Note that, with these assumptions,
e = C1x + D12u,
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H∞
H2
and the components, C1x and D12u are orthogonal. D12 is also assumed to be
normalized. This essentially means that there is no cross-weighting between the state
and input penalties. Assumption (iv) is the dual of this; the input and unknown input
(disturbance and noise) affect the measurement, y, orthogonally, with the weight on the
unknown input being unity.
To solve the H∞ design problem we define two Hamiltonian matrices,
ꢀ
ꢁ
A
γ
−2B1B1T − B2B2T
H
∞
=
,
−C1T C1
−AT
and
ꢀ
ꢁ
AT
−B1B1T
γ
−2C1T C1 − C2T C2
J
=
.
∞
−A
The following theorem gives the solution to the problem.
Theorem 3 There exists a stabilizing controller satisfying kG(s)k∞ < γ if and only if
the following three conditions are satisfied:
a) H ∈ dom(Ric) and X = Ric(H ) ≥ 0.
∞
∞
∞
b) J ∈ dom(Ric) and Y = Ric(J ) ≥ 0.
∞
∞
∞
c) ρ(X Y ) < γ2.
∞
∞
When these conditions are satisfied, one such controller is,
ꢀ
ꢁ
ˆ
F
A
−Z L
∞
∞
∞
∞
K (s) =
∞
,
0
where,
F
∞
=
−BT X
∞
2
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L
Z
=
=
=
−Y CT
∞
∞
∞
∞
2
(I − γ−2Y X )−1
∞
∞
A
A + γ−2B1BT X + B2F + Z L C2.
ˆ
∞
∞
∞
∞
1
Actually, the above formulation can be used to parametrize all stabilizing controllers
which satisfy, kG(s)k∞ < γ. This can be expressed as an LFT. All such controllers are
given by,
K
= Fl(M , Q),
∞
∞
where,
ˆ
A
∞
∞
−C2
−Z L
Z B2
∞
∞
∞
M
=
,
F
0
I
∞
I
0
and Q satisfies: Q ∈ RH∞, kQk∞ < γ. Note that if Q = 0 we get back the controller
given in Theorem 3. This controller is referred to as the central controller and it is the
controller calculated by the software.
Note also that the controller given above satisfies kGk∞ < γ. It is not necessarily the
controller that minimizes kGk∞ and is therefore referred to as a suboptimal H∞
controller. In practice this is not a problem, and may even be an advantage. The optimal
H∞ controller has properties which may not be desirable from an implementation point
of view. One typical property is that the high frequency gain is often large. Suboptimal
central controllers seem to be less likely to exhibit this characteristic.
2.3.5 Further Notes on the H∞ Design Algorithm
Now that we have covered the problem solution we can look at the areas that might give
potential numerical problems. The above results give a means of calculating a controller
(if one exists) for a specified γ value. As we mentioned earlier, the smallest such γ is
referred to as γopt. An iterative algorithm is used to find γ close to γopt and calculate
the resulting controller. The algorithm can be stated conceptually as follows:
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H∞
H2
a) Choose γ ≥ γopt
b) Form H and J
∞
∞
c) Check that H ∈ dom(Ric) and J ∈ dom(Ric).
∞
∞
d) Calculate X = Ric(H ) and Y = Ric(J )
∞
∞
∞
∞
e) Check that X ≥ 0 and Y ≥ 0
∞
∞
f) Check that ρ(X Y ) < γ2
∞
∞
g) Reduce γ and go to step b).
The value of γ can be reduced until one of the checks at steps c), e) or f) fails. In this
case, γ < γopt and we use the X and Y of the lowest successful γ calculation to
∞
∞
generate the controller. In the Xµ software a bisection search over γ is used to find a γ
close to γopt. If step a) is not satisfied, the routine exits immediately and tells the user
to select a higher initial choice for γ.
As part of the check that H ∈ dom(Ric), (and J ∈ dom(Ric)) the real part of the
∞
∞
eigenvalues is calculated. The software uses a tolerance to determine whether or not to
consider these zero. The default tolerance works well in most cases; the user can adjust
it if necessary.
In practice determining that X (and Y ) is positive definite involves checking that,
∞
∞
min Re{λi(X )} ≥ −ꢀ.
∞
i
Again, ꢀ is a preset tolerance which can be adjusted by the user if necessary.
The third check is that,
ρ(γ−2X Y ) < 1.
∞
∞
Fortunately this is a relatively well conditioned test.
The software displays the critical variables relating to each of these tests. The minimum
real part of the eigenvalues of H (and J ) is displayed. Similarly the minimum
∞
∞
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eigenvalue of X (and Y ) is displayed. The ultimate test of the software is to form the
∞
∞
closed loop system and check both its stability and norm. We strongly suggest that the
user always perform this step.
The numerical issues discussed above are very unlikely to arise in low order systems.
Experience has shown that systems with very lightly damped modes are more
susceptible to numerical problems than those with more heavily damped modes.
However, it has been found to be possible, with the software provided, to design
controllers using 60th order interconnection structures with very lightly damped modes.
2.3.6 H2 Design Overview
Recall from Section 2.1.2 that the H2 norm of a frequency domain transfer function,
G(s), is
ꢂ
ꢃ
Z
1/2
∞
1
2π
kG(s)k2 =
Trace [G(ω)∗G(ω)] dω
.
−∞
Several characterizations of this norm are possible in terms of input/output signals. For
example, if the unknown signals are of bounded energy, kGk2 gives the worst case
magnitude of the outputs e. Alternatively, if impulses are applied to the inputs of G(s),
kG(s)k2 gives the energy of the outputs e. H2 synthesis involves finding the controller
which minimizes the H2 norm of the closed loop system. This is the same the well
studied Linear Quadratic Gaussian problem.
2.3.7 Details of the H2 Design Procedure
The H2 design procedure is best explained by contrasting it with the H∞ procedure
explained in the previous sections. There are several differences, the most obvious being
that the H2 design problem always has a unique, minimizing, solution. The other
difference is that (in addition to the four conditions given in Section 2.3.2) D11 is
required to be zero, even in the general case. If this condition is violated no controller
will give a finite H2 norm for the closed loop system as it will not roll off as the
frequency goes to ∞.
We present the H2 solution in an LFT framework rather than the more well known LQG
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H∞
H2
framework. We again assume the simplifying assumptions used in Section 2.3.4 The H2
design solution is obtained (at least conceptually) from the H∞ design procedure by
setting γ = ∞ and using the resulting central controller. It is interesting to compare the
H∞ solution, given above, and the H2 solution given below.
Define two Hamiltonians, H2 and J2, by,
ꢀ
ꢁ
A
−B2B2T
H2 =
,
−C1T C1 −AT
and
ꢀ
ꢁ
AT
−B1B1T
−C2T C2
J2 =
.
−A
The sign definiteness of the off-diagonal blocks guarantees that H2 ∈ dom(Ric),
J2 ∈ dom(Ric) and X2 = Ric(H2) ≥ 0 and Y2 = Ric(J2) ≥ 0. The following theorem
gives the required result.
Theorem 4 The unique H2 optimal controller is given by,
ꢀ
ꢁ
ˆ
F2
A2 −L2
K2(s) =
,
0
where,
F2
L2
=
=
=
−B2T X2
−Y2C2T
ˆ
A2
A + B2F2 + L2C2.
We commented above that the controller, K2, is (conceptually) obtained by choosing
γ = ∞ in the H∞ design procedure. This does not mean that kGk∞ = ∞; it simply
means that we can make no a priori prediction about kGk∞ for this controller. K2
minimizes kGk2 and yields a finite kGk∞. As such, it is often useful for determining an
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Figure 2.8: Closed loop system, G(s), for performance analysis
initial choice of γ for the H∞ design procedure. We will see later (Section 2.5) that it
can also be used to initialize the D-K iteration procedure when an open-loop H∞ design
is poorly conditioned.
2.4 µ Analysis
2.4.1 Measures of Performance
Section 2.3 presented design performance objectives in terms of the norm (H2 or H∞) of
a closed loop system. We will now expand on this idea of performance. Consider the
closed loop system illustrated in Figure 2.8. The interconnection structure, P(s), is
specified such that w represents unknown inputs; typically reference commands,
disturbances and noise. The outputs, e, represent signals that we would like to be small.
“Small” means in the sense of a selected norm. These signals might include actuator
effort, and tracking error. As Figure 2.8 suggests, this analysis is typically applied after
calculating a controller.
The inputs w are described only as members of a set. The performance question is then:
For all w in this set, are all possible outputs e also in some set? The following set
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descriptions are considered, where B again denotes the unit ball.
(
(
(
)
ꢅ
ꢅ
ꢅ
ꢅ
Z
T
1
2T
2
Power :
BP =
w
w
w
lim
| w(t) | dt ≤ 1
(2.12)
(2.13)
(2.14)
T →∞
−T
)
ꢅ
ꢅ
ꢅ
ꢅ
Z
∞
2
2
Energy :
BL2 =
kwk2 =
| w(t) | dt ≤ 1
−∞
)
ꢅ
ꢅ
ꢅ
ꢅ
Magnitude : BL∞
=
kwk∞ = ess sup |w(t)| ≤ 1
t
These norms are defined for scalar signals for clarity. The choice of w and e as the above
sets defines the performance criterion. The performance can be considered as a test on
the corresponding induced norm of the system. More formally,
Lemma 5 (Nominal Performance)
For all w in the input set, e is in the output set
if and only if kG(s)k ≤ 1.
Only certain combinations of input and output sets lead to meaningful induced norms.
The H∞/µ approach is based on the cases w, e ∈ BP and w, e ∈ BL2. As we noted in
Section 2.1.2, both of these cases lead to the following induced norm.
kG(s)k∞ = sup σmax [G(ω)] .
ω
The choice of other input and output sets can lead to meaningful norms with
engineering significance. For example w, e ∈ BL∞ is arguably a more suitable choice for
some problems and leads to kGk1 as a performance measure where
Z
kGk1 = ∞|g(τ)| dτ.
0
and g(τ) is the convolution kernel (impulse response) of G(s). For a discussion on the
other possible selections of input and output sets, and the mathematical advantages of
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Figure 2.9: Perturbed closed loop system for stability analysis
the induced norms, the reader is referred to Doyle [2]. The major advantage of choosing
BP or BL2 is that the test for the performance can be considered in terms of the same
norm as stability. This has significant advantages when we are considering performance
and stability in the presence of perturbations, ∆.
2.4.2 Robust Stability and µ
Now we will consider the stability of a closed loop system under perturbations. In
Figure 2.9, G(s), is a perturbation model of a closed loop system. In the following
robust stability and robust performance analyses we will assume that ∆ is linear and
time-invariant.
We will also assume that the interconnection structure G(s) consists of stable transfer
function matrices, where stability is taken to mean that the system has no poles in the
closed right half plane. In practice this amounts to assuming that G22(s) (the nominal
closed loop system model) is stable as the other elements, G11(s), G12(s), and G21(s),
are weighting functions and can be chosen to be stable. The nominal closed loop system,
G22(s), often arises from a standard design procedure (H2, H∞, or loopshaping for
example) and will be stable.
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Consider the case where the model has only one full ∆ block (m = 1 and q = 0 in
Equation 2.9). This is often referred to as unstructured, and the well known result (refer
to Zames [67] and Doyle and Stein [5]) is given in the following lemma.
Lemma 6 (Robust Stability, Unstructured)
Fu(G(s), ∆) is stable for all ∆, k∆k∞ ≤ 1,
if and only if kG11(s)k∞ < 1.
A generalization of the above is required in order to handle Fu(G(s), ∆) models with
more than one full ∆ block. The positive real valued function µ can be defined on a
complex valued matrix M, by
det(I − M∆) = 0 for all ∆ ∈ B∆,
if and only if µ(M) < 1.
Note that µ scales linearly. In other words, for all α ∈ R,
µ(αM) = |α|µ(M).
In practice the test is normalized to one with the scaling being absorbed into the
interconnection structure. An alternative definition of µ is the following.
0 if no ∆ ∈ ∆ solves det(I + M∆) = 0
otherwise
µ(M) =
ꢀ
ꢁ−1
ꢅ
ꢅ
ꢅ
n
o
min
β
∃∆, k∆k ≤ β, such that det(I + M∆) = 0
∆∈∆
Note that µ is defined as the inverse of the size of the smallest destabilizing
perturbation. This immediately gives the following lemma.
Lemma 7 (Robust Stability, Structured)
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Fu(G(s), ∆) stable for all ∆ ∈ B∆
if and only if kµ(G11(s))k∞ < 1.
where
kµ(G11(s))k∞ = sup µ[G11(ω)].
ω
The use of this notation masks the fact that µ is also a function of the perturbation
structure, ∆. The above definition of µ applies to the more general block structure given
in Section 2.2.4. We can even consider the some of the blocks to be real valued, rather
than complex valued. The robust stability lemma is still valid; however the calculation
of µ becomes significantly more difficult.
In applying the matrix definition of µ to a real-rational G11(s), it has been assumed that
∆ is a complex constant at each frequency. This arises from the assumption that ∆ is
linear and time-invariant. Under this assumption we can examine the combination of
system and perturbation independently at each frequency. The analysis then involves
looking for the worst case frequency. If ∆ is not time-invariant then the frequency by
frequency analysis does not apply; ∆ can be used to shift energy between frequencies
and cause instability not predicted by the above analysis.
In practice this µ test is applied by selecting a frequency grid and at each frequency
calculating µ(G11(ω)). The choice of range and resolution for this grid is a matter of
engineering judgement. If very lightly damped modes are present a fine grid may be
required in the region of those modes.
2.4.3 Robust Performance
The obvious extension to the above is to consider performance in the presence of
perturbations ∆. For e, w ∈ BP or BL2 robust performance is a simple extension of
robust stability.
Lemma 8 (Robust Performance)
Fu(G(s), ∆) is stable and kFu(G(s), ∆)k∞ ≤ 1 for all ∆ ∈ B∆
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if and only if kµ(G(s))k∞ < 1,
b
where µ is taken with respect to an augmented structure ∆,
ꢅ
ꢅ
n
o
dim(w)×dim(e)
b
ˆ
ˆ
∆ = diag(∆, ∆) ∆ ∈ ∆, ∆ = C
.
ꢅ
ˆ
The additional perturbation block, ∆ can be thought of as a “performance block”
appended to the ∆ blocks used to model system uncertainty. This result is the major
benefit of the choice of input and output signal norms; the norm test for performance is
the same as that for stability. Robust performance is simply another µ test with one
additional full block.
The frequency domain robustness approach, outlined above, assumes that the
perturbations, ∆, are linear and time-invariant. This assumption is the most commonly
applied. Section 2.4.6 will consider a robustness analysis from a state-space point of
view. This form of analysis applies to norm bounded non-linear or time varying
perturbations. We will first look more closely at the properties of µ, particularly as they
relate to its calculation.
2.4.4 Properties of µ
The results presented here are due to Doyle [68]. Fan and Tits [69, 70] have done
extensive work on algorithms for tightening the bounds on the calculation of µ.
Packard [3] has also worked on improvement of the bounds and the extension of these
results to the repeated block cases. The most comprehensive article on the complex
singular value is that by Packard and Doyle [20]. More detail is contained in the
technical report by Doyle et al. [71].
We will look at simple bounds on µ. The upper bound results are particularly important
as they will form the basis of the design procedure provided in this software (D-K
iteration).
Defining a block structure made up of one repeated scalar, (∆ = {λI | λ ∈ C}) makes
the definition of µ the same as that of the spectral radius.
ꢅ
ꢅ
n
o
∆ = λI λ ∈ C
⇒
µ(M) = ρ(M).
ꢅ
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n×n
For the other extreme consider a single full block (∆ = {∆ | ∆ ∈ C
}); the definition
of µ is now the same as that for the maximum singular value,
n×n
∆ = {∆ | ∆ ∈ C
}
⇒
µ(M) = σmax(M).
Observe that every possible block structure, ∆, contains {λI | λ ∈ C} as a perturbation;
and every possible block structure, ∆, is contained in Cn×n. These particular block
structures are the boundary cases. This means that the resulting µ tests act as bounds
on µ for any block structure, ∆. This gives the following bounds.
ρ(M)
≤
µ(M)
≤
σmax(M).
The above bounds can be arbitrarily conservative but can be improved by using the
following transformations. Define the set
ꢅ
ꢅ
n
D = diag(D , . . . , D , d I , . . . , d I , ) D = D∗ > 0,
ꢅ
1
q
1
1
m
m
j
j
o
dim(Ii) = ki, di ∈ R, di > 0 (.2.15)
This is actually the set of invertible matrices that commute with all ∆ ∈ ∆. This allows
us to say that for all D ∈ D and for all ∆ ∈ ∆,
D
−1∆D = ∆.
Packard [3] shows that the restriction that di be positive real is without loss of
generality. We can actually take one of these blocks to be one (or the identity).
Now define Q as the set of unitary matrices contained in ∆:
ꢅ
n
o
ꢅ
Q = Q ∈ ∆ Q∗Q = I
.
(2.16)
ꢅ
The sets D and Q can be used to tighten the bounds on µ in the following way (refer to
Doyle [68]).
max ρ(QM)
≤
µ(M)
≤
inf σmax(DMD−1).
(2.17)
Q∈Q
D∈D
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Actually, the lower bound is always equal to µ but the implied optimization has local
maxima which are not global. For the upper bound Safonov and Doyle [72], have shown
that finding the infimum is a convex problem and hence more easily solved. However the
bound is equal to µ only in certain special cases. Here we use the infimum rather that
the miminum because D may have a element which goes to zero as the maximum
singular value decreases. So the limiting case (where an element of D is zero) is not a
member of the set D.
The cases where the upper bound is equal to µ are tabulated below.
q = 0
q = 1
q = 2
m = 0
m = 1
m = 2
m = 3
m = 4
equal
equal
less than or equal
less than or equal
equal
equal
equal
less than or equal less than or equal
less than or equal less than or equal
less than or equal less than or equal less than or equal
Most practical applications of the µ theory involve models where q = 0. Here we see that
we have equality with the upper bound for three or fewer blocks. Computational
experience has yet to produce an example where the bound differs by more than
15 percent. In practically motivated problems the gap is usually much less.
2.4.5 The Main Loop Theorem
We will introduce a fundamental theorem in µ analysis: the main loop theorem. From
the previous discussion you will see that there are several matrix properties that can be
expressed as µ tests. The spectral radius and the maximum singular value are two such
quantities. The main loop theorem gives a way of testing such properties for perturbed
systems. The test is simply a larger µ problem. This is the theorem underlying the
extension from robust stability to robust performance.
Consider a partitioned matrix,
ꢀ
ꢁ
M11 M12
M21 M22
M =
,
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and two block structures, ∆1 (compatible with M11) and ∆2 (compatible with M22).
There are two perturbed subsystems that we can study here: Fu(M, ∆1), where ∆1 is
closed in a feedback loop around M11; and Fl(M, ∆2), where ∆2 is closed in a feedback
loop around M22.
We have already seen that in the case of a dynamic system, the robust stability of
Fu(M, ∆1) is analyzed by checking that µ1(M11) < 1. Here we have used µ1 to indicate
that we are considering it with respect to the block structure ∆1. In the constant
matrix case, we say that the LFT, Fu(M, ∆1) is well posed for all ∆1 ∈ B∆1 if and only
if µ1(M11) < 1. This simply means that the inverse in the LFT equations is well defined
for all ∆1 ∈ B∆1.
The well posedness discussion above applies equally well to Fl(M, ∆2) and we will
denote the µ test for M22 by µ2(M22). However, instead of looking at µ2 of M22, we
want to look at µ2(Fu(M, ∆1)). Note that Fu(M, ∆1) has the same dimensions as M22
and in fact Fu(M, ∆1) = M22 when ∆1 = 0. In otherwords, what happens when we
apply the µ2 test to the whole set of matrices generated by Fu(M, ∆1).
To answer this question we need to introduce a larger block structure, denoted here
simply by ∆. This is simply the diagonal combination of the previous two structures:
∆ = diag(∆1, ∆2).
Note that this has compatible dimensions with M itself and the associated µ test will be
denoted by µ(M). Now we can answer the question about what happens to
µ2(Fu(M, ∆1)) for all ∆1 ∈ B∆1.
Theorem 9 (Main Loop Theorem)
µ1(M11) < 1
and
µ(M) < 1
if and only if
max µ2[Fu(M, ∆1)] < 1
∆1∈B∆1
This theorem underlies the fact that robust performance is a simple extension of robust
stability. It has a much more significant role in developing connections between the µ
theory and other theoretical aspects of control. The example in the following section is
an illustration of this point.
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2.4.6 State-space Robustness Analysis Tests
We will look at some more advanced state-space approaches to the analysis of robust
performance. Most users of the software will concentrate on the more common frequency
domain analysis methods covered in Section 2.4.3. The analysis tests given here can be
implemented with the Xµ functions and the more advanced user can use these to study
the robustness of systems with respect to time-varying and nonlinear perturbations.
To illustrate this approach, consider the state-space LFT model of a system introduced
in Section 2.2.4. A digital system is described by,
x(k + 1)
=
=
Ax(k) + Bu(k)
y(k)
Cx(k) + Du(k).
This digital system has transfer function,
P(z) = Fu(Pss, z−1I),
where Pss is the real valued matrix,
ꢀ
ꢁ
A B
C D
Pss
=
,
and the scalar × identity, z−1I, has dimension nx, equal to the state dimension of P(z).
This is now in the form of an LFT model with a single scalar × identity element in the
upper loop.
Define,
ꢅ
ꢅ
ꢉ
∆1 = δInx δ ∈ C
,
and note that with this definition,
µ1(A) = ρ(A).
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Therefore µ1(A) < 1 is equivalent to our system being stable. Furthermore, the
maximum modulus theorem for a stable system tells us that,
kP(z)k∞
=
=
=
|szu|≥p1 σmax(P(z))
sup σmax(Fu(Pss, z−1I)
|z−1|≤1
sup µ2(Fu(Pss, ∆1),
∆1∈B∆1
if ∆2 is defined as a single full block of dimensions matching the input-output
dimensions of P(z). The main loop theorem (Theorem 9) immediately suggests the
following result.
Lemma 10
P(z) is stable
and
kP(z)k∞ < 1.
µ(Pss) < 1
if and only if
Note that this tests whether or not the ∞ norm is less than one. It doesn’t actually
calculate the ∞ norm. To do this we have to set up a scaled system and search over the
scaling with makes µ(Pss) = 1.
To apply this to a robust performance problem, consider the configuration shown in
Figure 2.10. This is a state-space representation of a perturbed system. It would
typically model an uncertain closed-loop system where the performance objective is
kek ≤ kwk, for all w ∈ BL2 and all ∆ ∈ B∆.
The real-valued matrix, Gss, is,
A
B1
B2
Gss
=
C1 D11 D12
C2 D21 D22
.
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Figure 2.10: Perturbed system for state-space robustness tests
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Note that the nominal system is given by,
ꢂꢀ
ꢁ
ꢃ
A B1
C1 D11
Gnom(z) = Fu
, z−1I
,
and the perturbed system is,
G(z) = Fu(Fl(G, ∆), z−1I).
We assume that ∆ is an element of a unity norm bounded block structure, ∆ ∈ B∆.
For the µ analysis we will define a block structure corresponding to Gss,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
∆s = diag(δ1Inx, ∆2, ∆) δ1 ∈ C, ∆2 ∈ Cdim(w)×dim(e), ∆ ∈ ∆
.
ꢅ
Consider also a block structure corresponding to Fu(Gss, z−1I),
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
∆p = diag(∆2, ∆) ∆2 ∈ Cdim(w)×dim(e), ∆ ∈ ∆
.
ꢅ
This is identical to the ∆s structure except that the δ1Inx block, corresponding to the
state equation, is not present. The following theorem gives the equivalence between the
standard frequency domain µ test and a state-space µ test for robust performance (first
introduced by Doyle and Packard [23]. The notation µ∆ will denote a µ test with
s
respect to the structure ∆s, and µ∆ is a µ test with respect to the ∆p structure.
p
Theorem 11
The following conditions are equivalent.
i) µ∆ (Gss) < 1
(state-space µ test);
s
ii) ρ(A) < 1 and max µ∆ (Fu(Gss, eωI)) < 1
(frequency domain µ test);
p
ω∈[0,2π]
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iii) There exists a constant β ∈ [0, 1] such that for each fixed ∆ ∈ B∆, G(z) is stable
and for zero initial state response, e satisfies kek2 ≤ β kwk2
(robust
performance).
The frequency domain µ test is implemented by searching for the maximum value of µ
over a user specified frequency grid. Theorem 11 shows that this is equivalent to a
single, larger, µ test. There are subtle distinctions between the two tests. As we would
expect, calculation of the larger µ test is more difficult. More importantly, the result
does not scale. In the frequency domain test, if
ω
max µ∆ [Fu(Gss, e
I
nx
)] = β,
p
ω∈[0,2π]
where β > 1, then we are robust with respect to perturbations up to size 1/β. In the
state-space test, if µ∆ (Gss) = β, where β > 1, then we cannot draw any conclusions
s
about the robust performance characteristics of the system. We must scale the inputs or
outputs and repeat the calculation until the µ test gives a result less than one.
In practice we can only calculate upper and lower bounds for both of these µ tests.
Although the state-space and frequency domain µ tests are equivalent, their upper
bound tests have different meanings. We will see that this difference can be used to
study the difference between linear time-invariant perturbations and linear time-varying
(and some classes of non-linear) perturbations.
To clarify this issue, consider the D scales which correspond to ∆s and ∆p;
ꢅ
ꢅ
ꢅ
ꢄ
T
Ds
Dp
=
=
diag(D1, d2I2, D) D1 = D1 > 0, dim(D1) = nx × nx,
ꢅ
ꢆ
d2 > 0, dim(I2) = dim(w) × dim(w), D ∈ D
,
ꢅ
ꢄ
ꢆ
ꢅ
ꢅ
diag(d2I2, D) d2 > 0, dim(I2) = dim(w) × dim(w), D ∈ D
.
ꢅ
In the above D is the set of D-scales for the perturbation structure ∆, and, for
notational simplicity, we have assumed that dim(w) = dim(e). Now, the upper bound
tests are:
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i) State-space upper bound:
inf σmax[DsGssDs−1] < 1;
Ds∈Ds
ii) Frequency domain, constant D, upper bound:
ω
inf
max σmax[DpFu(Gss, e
I
nx
)D−1] < 1;
p
Dp∈D
p ω∈[0,2π]
iii) Frequency domain upper bound:
ω
max inf σmax[DpFu(Gss, e
I
nx
)D−1] < 1.
p
D ∈D
ω∈[0,2π]
p
p
In both the state-space and frequency domain, constant D, upper bound, a single D
scale is selected to guarantee robust performance over all frequencies. These two tests
(items i) and ii) above) are equivalent. In the frequency domain upper bound test
(item iii)) a different D-scale is selected at each frequency.
The relationship between all of these tests is summarized by the following:
inf σmax[DsGssDs−1] < 1
State-space upper bound
Ds∈Ds
m
ω
ω
inf
max σmax[DpFu(Gss, e
I
)D−1] < 1
Frequency domain, constant D,
upper bound
nx
nx
p
Dp∈D
p ω∈[0,2π]
⇓
max inf σmax[DpFu(Gss, e
I
)D−1] < 1
Frequency domain upper bound
Frequency domain µ test
State-space µ test
p
D ∈D
ω∈[0,2π]
p
p
⇓
ω
max µ∆ [Fu(Gss, e
I
nx
)] < 1
p
ω∈[0,2π]
m
µ∆ [Gss] < 1
s
In the two cases where there are one way implications, there are real gaps. We have
already seen that there is a gap between the frequency domain µ test and its upper
bound for four or more full blocks. This is a computational issue.
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The gap between the state-space (or constant D) upper bound and the frequency
domain upper bound is more significant. In the state-space upper bound, a single D
scale is selected. This gives robust performance for all ∆ satisfying, kvk ≤ kzk for all
e ∈ L2. This can be satisfied for linear time-varying perturbations or non-linear cone
bounded perturbations. The formal result is given in the following theorem (given in
Packard and Doyle [20]).
Theorem 12
If there exists Ds ∈ Ds such that
σmax[DsGssDs−1] = β < 1,
then there exists constants, c1 ≥ c2 > 0 such that for all perturbation sequences,
{∆(k)}∞k=0 with ∆(k) ∈ ∆, σmax[∆(k)] < 1/β, the time varying uncertain system,
ꢀ
ꢁ
ꢀ
ꢁ
x(k + 1)
e(k)
x(k)
w(k)
= Fl(Gss, ∆(k))
,
is zero-input, exponentially stable, and furthermore if {w(k)}∞k=0 ∈ l2, then
2
2
2
2
c2(1 − β2) kxk2 + kek2 ≤ β2 kwk2 + c1kx(0)k .
In particular,
2
2
2
kek2 ≤ β2 kwk2 + c1kx(0)k .
The user now has a choice of robust performance tests to apply. The most appropriate
depends on the assumed nature of the perturbations. If the state-space upper bound test
is used, the class of allowable perturbations is now very much larger and includes
perturbations with arbitrarily fast time variation. If the actual uncertainty were best
modeled by a linear time-invariant perturbation then the state-space µ test could be
conservative. The frequency domain upper bound is probably the most commonly used
test. Even though the uncertainties in a true physical system will not be linear, this
assumption gives suitable analysis results in a wide range of practical examples.
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2.4.7 Analysis with both Real and Complex Perturbations
The above results only apply to the case where ∆ is considered as a constant complex
valued matrix at each frequency. In many engineering applications restricting certain of
the ∆ blocks to be real valued may result in a less conservative model. Analysis with
such restrictions is referred to as the “mixed” µ problem.
For example, consider the LFT form of the engine combustion model developed in
Section 2.2.4 (Equation 2.10). The block structure contains both real and complex
perturbations. A closed-loop model will also include these perturbations and the robust
stability and robust performance analyses will involve calculation of µ with respect to
both real and complex valued perturbations. We could simply assume that all
perturbations were complex; this would certainly cover the situation. However, such an
assumption may be too conservative to be useful. Calculation of mixed µ will give a
more accurate result in this case.
Efficient computation of µ in the mixed case is discussed by Doyle, Fan, Young, Dahleh
and others [73, 74, 75, 76]. Accurate mixed µ analysis software will be available in the
near future. Unlike the complex µ case, this will not directly lead to a compatible
synthesis procedure. Significantly more work is required in this direction.
2.5 µ Synthesis and D-K Iteration
2.5.1 µ-Synthesis
We now look at the problem of designing a controller to achieve a performance
specification for all plants, P(s), in a set of plants, P. The previous sections have dealt
with the questions of performance and robust stability in depth and the same framework
is considered for the synthesis problem. Figure 2.11 illustrates the generic synthesis
interconnection structure.
The lower half of this figure is the same as that for the H∞ and H2 design procedure.
The controller measurements are y, and the controller actuation inputs to the system are
u. The configuration differs from the standard H∞ or H2 case in that Fu(P(s), ∆)
(rather than the nominal plant, P22(s)) is used as the design interconnection structure.
The problem is to find K(s) such that for all ∆ ∈ B∆, K(s) stabilizes Fu(P(s), ∆) and
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Figure 2.11: The generic interconnection structure for synthesis
kFu(Fl(P(s), K(s)), ∆)k∞ ≤ 1. This is equivalent to K(s) satisfying
µ[Fl(P(s), K(s))] < 1. In other words, the closed loop system satisfies the robust
performance specification.
Unfortunately this problem has not yet been solved, except is a few special cases. The
current approach to this problem, known as D-K iteration, involves the iterative
application of the H∞ design technique and the upper bound µ calculation. We will give
a brief conceptual overview here and give more algorithmic details in Section 2.5.2.
Consider applying H∞ synthesis to the full P(s) interconnection structure for this
problem. Suppose that this gives a controller, K(s) such that K(s) stabilizes P(s) and
kFl(P(s), K(s))k∞ ≤ 1.
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Recall that this is an upper bound for the µ problem of interest, implying that,
µ[Fl(P(s), K(s)] ≤ 1,
as required. However the upper bound may be conservative, meaning that in order to
guarantee that µ[Fl(P(s), K(s)] ≤ 1, we have had to back off on the performance and/or
the stability margins.
With the appropriate choice of D scalings the upper bound will be much closer to µ. In
otherwords there exists D such that, kDFl(P(s), K(s))Dk∞ is a close upper bound to
µ[Fl(P(s), K(s)]. The µ-synthesis problem can be replaced with the following
approximation (based on the upper bound):
inf
D∈D
kDFl(P(s), K(s))D−1k∞.
(2.18)
K(s) stabilizing
The reader is referred to Doyle [1] for details of this problem.
If this is considered as an optimization of two variables, D and K(s), the problem is
convex in each of the variables separately, but not jointly convex. Doyle [2] gives an
example where this method reaches a local nonglobal minimum.
D-K iteration involves iterating between using D ∈ D and K(s) to solve Equation 2.18.
There are several practical issues to be addressed in doing this and we discuss those in
the next section.
2.5.2 The D-K Iteration Algorithm
The objective is to design a controller which minimizes the upper bound to µ for the
closed loop system;
inf
D∈D
kDFl(P(s), K(s))D−1k∞.
K(s) stabilizing
The major problem in doing this is that the D-scale that results from the µ calculation
is in the form of frequency by frequency data and the D-scale required above must be a
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dynamic system. This requires fitting an approximation to the upper bound D-scale in
the iteration. We will now look at this issue more closely.
The D-K iteration procedure is illustrated schematically in Figure 2.12. It can be
summarized as follows:
i) Initialize procedure with K0(s): H∞ (or other) controller for P(s).
ii) Calculate resulting closed loop: Fl(P(s), K(s)).
iii) Calculate D scales for µ upper bound:
inf σmax[D(ω)Fl(P(s), K(s))D(ω)−1].
D(ω)∈D
ˆ
ˆ
iv) Approximate frequency data, D(ω), by D(s) ∈ RH∞, with D(ω) ≈ D(ω).
v) Design H∞ controller for D(s)P(s)D−1(s).
ˆ
ˆ
vi) Go to step ii).
We have used the notation D(ω) to emphasize that the D scale arises from frequency by
frequency µ analyses of G(ω) = Fl(P(ω), K(ω)) and is therefore a function of ω. Note
that it is NOT the frequency response of some transfer function and therefore we do
NOT use the notation D(ω).
The µ analysis of the closed loop system is unaffected by the D-scales. However the H∞
design problem is strongly affected by scaling. The procedure aims at finding at D such
that the upper bound for the closed loop system is a close approximation to µ for the
closed loop system. There are several details about this procedure that will now be
clarified.
At each frequency, a scaling matrix, D(ω), can be found such that
σmax(D(ω)G(ω)D(ω)−1) is a close upper bound to µ(G(ω)) (Figure 2.12c). The D
scale is block diagonal and the block corresponding to the e and w signals can be chosen
to be the identity. The part of D corresponding to the z signal commutes with ∆ and
cancels out the part of D−1 corresponding to the v signal. To illustrate this, consider
the D scale that might result from a block structure with only m full blocks. At each
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Figure 2.12: D-K iteration procedure: a) Design H∞ (or other) controller: K0(s) [step
i)]. b) Closed loop perturbed system for µ analysis [step ii)]. c) Frequency by frequency
upper bound D(ω) scale approximation to µ analysis [step iii)]. d) Scaling of H∞ design
ˆ
ˆ
problem by D(s) where D(ω) ≈ D(ω) [steps iv) & v)].
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frequency we would have,
d1I1
.
.
.
D =
,
dmIm
Ie
where the identity Ie is of dimensions dim(e) × dim(e).
The calculation of a new H∞ controller requires a state-space realization of D(ω). For
each di in D(ω) we must fit a transfer function approximation, which we will denote by
ˆ
ˆ
di(s). This is denoted by D(s) in the above discussion. The observant reader will notice
ˆ
that, as defined here, D(s) is not of the correct input dimension to multiply P(s). We
must append another identity of dimension equal to the dimension of the signal y. The
final result is,
ˆ
d1(s)I1
.
.
.
ˆ
D(s) =
ˆ
dm(s)Im
Ie
Iy
and
d−1 1(s)I1
ˆ
.
.
.
−1(s) =
.
ˆ
D
d−m1(s)Im
ˆ
Iw
Iu
Throughout the theoretical discussion we have assumed that the perturbation blocks,
∆i, were square. The software handles the non-square case. This makes a difference to
D(s) and D−1(s). The identity blocks (Im, etc.) shown above will be of different sizes
ˆ
ˆ
for D(s) and D−1(s) if the corresponding ∆i perturbation is non-square. Similarly, the
ˆ
ˆ
−1
ˆ
ˆ
Iw and Iu identities in D (s) are not necessarily the same size as Ie and Iy in D(s).
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Several aspects of this procedure are worth noting. For the µ analysis and D scale
calculation, a frequency grid must be chosen. The range and resolution of this grid is a
matter of engineering judgement. The µ analysis can require a fine grid in the vicinity of
the lightly damped modes. The order of the initial controller, K0(s), is the same as the
interconnection structure, G(s). The order of K(s) is equal to the sum of the orders of
G(s), D(s) and D−1(s). This leads to a trade-off between the accuracy of the fit
ˆ
ˆ
ˆ
between D and D(s) and the order of the resulting controller, K(s).
Another aspect of this to consider is that as the iteration approaches the optimal µ
value, the resulting controllers often have more and more response at high frequencies.
This may not show up in the µ calculation, the D scale fitting, or a frequency response
of K(s), because the dynamics are occuring outside of the user specified frequency grid.
However these dynamics affect the next H∞ design step and may even lead to numerical
instability.
The above discussion used an H∞ controller to initialize the iteration. Actually any
stabilizing controller can be used. In high order, lightly damped, interconnection
structures, the H∞ design of K0(s) may be badly conditioned. In such a case the
software may fail to generate a controller, or may give controller which doesn’t stabilize
the system. A different controller (the H2 controller is often a good choice) can be used
to get a stable closed loop system, and thereby obtain D scales. Application of these D
scales (provided that they do not add significantly many extra states) often results in a
better conditioned H∞ design problem and the iteration can proceed.
The robust performance difference between the H∞ controller, K0(s), and K(s), can be
dramatic even after a single D-K iteration. The H∞ problem is sensitive to the relative
scalings between v and w (and z and e). The D scale provides the significantly better
choice of relative scalings for closed loop robust performance. Even the application of a
constant D scale can have dramatic benefits.
2.6 Model Reduction
High order interconnection structures will result in high order controllers. Often a
controller of significantly lower order will perform almost as well. Approximating a
state-space system by one of lower order is referred to as model reduction. There are
several techniques available for this purpose in Xµ and the background to these
techniques is discussed here.
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2.6.1 Truncation and Residualization
The simplest form of model reduction is state truncation. Consider a system, P(s), with
a partitioned state matrix,
A11 A12 B1
P(s) = A21 A22 B2
.
C1 C2
D
Truncating the states associated with A22 results in,
ꢀ
ꢁ
A11 B1
Ptrun(s) =
.
C1
D
In any practical application we would order the states so that those truncated do not
significantly affect the system response. For example, to truncate high frequency modes,
A is transformed to be diagonal (or with 2 × 2 blocks on the diagonal) and the
eigenvalues are ordered in increasing magnitude. This results in A21 = 0 and A22
corresponds to the high frequency modes.
Truncation also affects the zero frequency response of the system. Residualization
involves truncating the system and adding a matrix to the D matrix so that the zero
frequency gain is unchanged. This typically gives a closer approximation to the original
system at low frequency. If the original system rolls off with frequency, the low order
residualized approximation will usually not share this characteristic. Using the above
P(s), the result is,
ꢀ
ꢁ
−1
A11 − A12A22
A
B1 − A12A−221B2
21
Presid(s) =
.
−1
C1 − C2A22
A
D − C2A−221B2
21
2.6.2 Balanced Truncation
Consider a stable state-space system,
ꢀ
ꢁ
A B
C D
P(s) =
.
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The controllability grammian, Y is defined as,
Z
∞
T
Y =
e
AtBBT eA tdt,
0
and the observability grammian, X, is defined as
Z
∞
T
X =
eA tCT CeAtdt.
0
The grammians, X and Y , satisfy the Lyapunov equations,
AY + Y AT + BBT
AT X + XA + CT C
=
=
0
0,
and this is typically how they are calculated. We can also see from the definitions that
X ≥ 0 and Y ≥ 0. Actually Y > 0 if and only if (A,B) is controllable and X > 0 if and
only if (C, A) is observable.
Now consider the effect of a state transformation on these grammians. Define a new
state, xˆ, by xˆ = T x, where T is invertible, to give
ꢀ
ꢀ
ꢁ
ˆ ˆ
A B
P(s)
=
=
ˆ
C D
ꢁ
TAT −1 TB
.
CT −1
D
T
The new grammians are Y = TY T and X = T −T XT −1. The product of the
ˆ
ˆ
ˆ ˆ
grammians, Y X is therefore given by,
−1
ˆ ˆ
Y X = T Y XT
.
This illustrates that the eigenvalues of the product of the grammians is invariant under
state similarity transformation.
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We will now look at a particular choice of transformation. For a minimal realization, we
can always find a transformation that gives,
T
ˆ
Y = T Y T = Σ,
and
X = T −T XT −1 = Σ,
ˆ
where Σ = diag(σ1, . . . , σn) and σi ≥ 0, i = 1, . . . , n. This realization, where the
grammians are equal, is called a balanced realization. Each mode of the balanced system
can be thought of as equally controllable and observable. Balanced realization was first
introduced by Moore [77].
The σi are known as the Hankel singular values of the system and are ordered such that
σ1 is the largest and σn is the smallest. Because the eigenvalues of the product of the
grammians are invariant with respect to similarity transformations, the Hankel singular
values are system invariants. We will denote the Hankel norm of a system as kP(s)kH
and this is given by,
kP(s)kH = σ1.
The input-output interpretation of the Hankel norm is the following,
ꢌ
ꢌ
ꢌ
ꢌ
ꢌ
ꢌ
ꢅ
ꢅ
y(t)
(0,∞)
2
kPkH
=
sup
.
ku(t)k2
u(t)∈L2(−∞,0)
ꢅ
ꢅ
The notation, y(t) (0,∞), denotes the system output, considered only over the time
interval zero to ∞. So we are looking at the system output, from time zero to ∞, in
response to input signals from −∞ to zero. The Hankel norm is the maximum gain from
past inputs to future outputs. Each signal, u(t) ∈ L2(−∞, 0) drives the system state to
a particular location in the state-space, and the output (considered over (0, ∞)) is the
corresponding transient decay from that state.
Balanced truncation involves obtaining a balanced realization of P(s) and then
truncating the states corresponding to the smallest Hankel singular values. Enns [78]
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and Glover [79] independently obtained the following bound on the error induced by
balanced truncation.
Theorem 13
Given a stable, rational, P(s), and Pbal(s), the balanced truncation of order k < n. Then,
n
X
kP(s) − Pbal(s)k∞
≤
≤
2
2
σi
i=k+1
and
n
X
kP(s) − Pbal(s)kH
σi.
i=k+1
Unobservable or uncontrollable modes have a corresponding Hankel singular value of
zero and we can see immediately from the above that their truncation does not affect
the ∞-norm of the system.
2.6.3 Hankel Norm Approximation
We can also consider the problem of finding the kth order controller which gives the
closest fit (in terms of the Hankel norm) to the original system. The results given here
are due to Glover [79]. The first thing to consider is a lower bound on the error, which is
specified in the following lemma.
Lemma 14
Given a stable, rational P(s), and a kth order approximation, Pk(s). Then
σk+1 ≤ kP(s) − Pk(s)k∞.
This tells us how well we can expect to do in terms of the ∞ norm. Actually, there
exists a Pk(s) which achieves this bound. The only problem is that it can have unstable
(or anti-causal) parts to it.
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Consider the problem of finding the stable, order k realization which minimizes the
Hankel norm of the error. Define, Phankel(s) as the minimizing system. Then we have,
σk+1
≤
kP(s) − Phankel(s)kH
=
inf
kP(s) − Pk(s)kH.
Pk(s) stable
This system also satisfies ∞-norm bounds on the error, as illustrated in the following
theorem.
Theorem 15
Given a stable, rational, P(s), and the optimal kth order Hankel norm approximation,
Phankel(s). Then
n
X
kP(s) − Phankel(s)k∞ ≤ 2
σi.
i=k+1
Furthermore, there exists a constant matrix, D0, such that
n
X
kP(s) − (Phankel(s) + D0)k∞
≤
σi.
i=k+1
Careful examination of the previous section will indicate that the Hankel norm of a
system is independent of the D term. The optimal Hankel norm approximation given
above, Phankel(s), is considered to have a zero D term. It has the same error bounds as
the balanced truncation. Theorem 15 states that we can find a D matrix to add to
Phankel(s) to cut this bound in half.
The most common use of balanced truncations and Hankel norm approximations is to
reduce the order of a controller. Note that this will give a small ∞-norm error with
respect to the open-loop controller. It does not say anything about the preservation of
closed loop properties. These should always be checked after performing a controller
order reduction.
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Chapter 3
Functional Description of Xµ
3.1 Introduction
This chapter describes the Xµ functions in the context of their intended usage.
Chapter 2 provides the reader with an idea of the theoretical basis behind the various
analysis and design calculations. Here we outline the software functions available for
doing those calculations.
Robust control design uses a subset of data objects provided within Xmath. We discuss
the details of the most heavily used objects: Dynamic Systems and pdms. This
coverage overlaps that given in the Xmath Basics manual; only the emphasis is different.
There are several subtleties which arise when using these data objects in a robust
control context. These issues are discussed in Section 3.2.
3.2 Data Objects
Xmath provides a wide range of data objects. There are several which are of primary
interest in control design: matrices, pdms and Dynamic System. The transfer function
object is useful for specifying systems although all calculations will be done with state
space Dynamic Systems. The control uses of these objects is reviewed in this section.
71
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3.2.1 Dynamic Systems
Xmath has a dynamic system data object which specifies a dynamic system in terms of
A, B, C and D matrices. The dynamic equations of the system are,
x˙(t)
=
=
Ax(t) + Bu(t),
y(t)
Cx(t) + Du(t),
in the continuous case, and
x(kT + T )
=
=
Ax(kT ) + Bu(kT ),
y(kT )
Cx(kT ) + Du(kT ),
in the discrete time case. The discrete time sample period, T , is stored as part of the
data object. The user can label the system inputs, u(t), outputs, y(t), and states, x(t).
Also residing within the Xmath state-space object is the initial state, x(0). This is used
primarily for time response calculations. It is debatable whether or not the initial state
is an intrinsic attribute of the system as one frequently changes it for simulation. It has
the advantage of reducing the time response calculation to a simple multiplication and it
can easily be changed without accessing all of the other system variables.
The Xmath core functions system and abcd are used to create state-space systems. The
system function is also used to specify the initial state (and, as discussed in the next
section, any other system attributes). The following example puts together a simple
two-state system.
# Specify the A, B, C & D matrices
a = [-.15,.5;-.5,-.15]
b = [.2,4;-.4,0]
c = [5,5]
d = [.1,-.1]
sys = system(a,b,c,d)
[a,b,c,d] = abcd(sys)
Simple systems are easily generated as transfer functions. To achieve this we simply
define the numerator and denominator polynomials and then divide one by the other.
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As above, these polynomials can be specified by their roots or their coefficients. Note
that we can specify the variable, and for continuous systems we use “s”. To create a
discrete system “z” is used.
# Generate the system from the numerator and denominator
# coefficients.
numerator = makepoly([-.1,19.97,-7.02725],"s");
denominator = makepoly([1,0.3,0.2725],"s");
# We can also do this by specifying the roots of each
# polynomial
numerator = -0.1*polynomial([199.3475;0.3525],"s");
denominator = polynomial( ...
[-.15+0.5*jay;-.15-0.5*jay],"s");
# Note that multiplying the by -0.1 does the correct
# thing to the polynomial above. The final system is
# obtained with the command:
sys1 = numerator/denominator
Labeling and Comments
Xmath allows the user to label all the inputs, outputs and states in a state-space system.
Any Xmath variable can also have a comment string associated with it.
Keywords for the system function are used to label the system. In the following
example, the two-input, single-output system generated above is used as the starting
point. Comments can be attached to any variable in the workspace (or the workspace
itself) with the comment command.
# Set up vectors of strings for the labels
inputs = ["disturbance";"actuator"]
outputs = ["measurement"]
states = ["x1";"x2"]
# and attach them to sys
sys = system(sys,inputNames = inputs,...
outputNames = outputs,...
stateNames = states)
# We can also attach a comment to sys
comment sys "Example system for the manual"
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Because the dynamic system is a built-in data object, the label information will be
displayed by simply typing the object name (sys in the above) or appending a question
mark to the statement. The core function commentof is used to read the comment
attached to a specified variable.
3.2.2 pdms
A pdm can be thought of as a three-dimensional matrix. The typical control uses are
time or frequency responses, where the domain variable (the third dimension) is time or
frequency. The frequency response of a multiple-input, multiple-output (MIMO) system
is a complex valued matrix at each frequency. A time response of a multiple-output
system is a real valued vector at each time step.
As we would expect, there is a core function for constructing pdms from the raw data:
pdm. The data matrices can be either bottom or right concatenated. If there is an
ambiguity, Xmath assumes that the data was bottom concatenated. Keywords can be
used to specify the row and column dimensions of the data matrices.
To extract the data again, the function makematrix returns the matrix data in right
concatenated form. The domain of a pdm is obtained with the function domain.
# Make a pdm with 2 values in the domain
mat1 = random(2,2)
mat2 = -1*random(2,2)
dom = [1;1+pi]
pdm1 = pdm([mat1;mat2],dom)
# Extract back the data and the domain
dom = domain(pdm1)
data = makematrix(pdm1)
The pdm data object also contains row, column and domain labels. These can be
appended with the pdm command in exactly the same manner as the Dynamic System
labels illustrated above. Refer to the Xmath Basics manual for graphical illustration and
further details about pdms.
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Appending and Merging Data
Time functions, for creating simulation inputs for example, can be created by combining
pdms. Xmath has two such core functions: concatseg and insertSeg The concatseg
appends the data of one pdm to another. The domain is recalculated such that it is
always regular. The user can specify a new beginning value for the domain and the
spacing is determined by a series of prioritized rules. insertSeg inserts the data from
one pdm into another. It can also “insert” before the beginning, or after the end, of the
first pdm and the resulting gap is filled with zeros. Again the domain of the result is
recalculated from a user specified beginning and is always regular. Both of these
functions are useful in creating time domain waveforms.
In the typical robust control cases where we want to merge pdm data, the result will not
have a regular domain. One example is merging experimental transfer function estimates
from multiple experiments, prior to performing a least squares transfer function fit or
error analysis. For this purpose, Xµ provides a mergeseg function. Keywords allow the
user to specify whether to sort in ascending or descending order or even whether to sort
at all. The following example illustrates the a typical use.
# pdm1 and pdm2 have identical row and column
# dimensions and at least some non matching
# independent variables
pdm3 = mergeseg(pdm1,pdm2,{ascending,!duplicates})
As an aside, note that the sort function in Xmath sorts each column of each matrix in a
pdm, rather than sorting the domain.
Extracting Data by Independent Variable/Domain
It is often useful to be able to extract a segment of matrix data from a pdm. The
function extractseg performs this function. The user has the option of resetting the
domain to a new starting value.
# sys1g is a freq. response from 0.1 to 100 rad/sec
# select data from 1 to 10 rad/sec as sys1g1
sys1g1 = extractseg(sys1g,1,10)
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Data can also be extracted by independent variable index number, by providing a scalar
argument to the pdm. In the following example the fifth through tenth and the
twentieth independent variables are specified for the smaller pdm, pdm2.
size(pdm1)
ans (a row vector) =
pdm2 = pdm1([5:10,20])
size(pdm2)
1
1
1
1
100
7
ans (a row vector) =
Indexing and Finding Data
The core function find returns the indices of the elements of a matrix which meet a user
defined criterion. By using a Boolean function of a matrix as the argument to find, the
an index to the entries satisfying the Boolean function can be generated. This function
works equally well for for pdms.
This is illustrated in the following example.
# sys3g is a MIMO frequency response.
# We want to find the frequencies where the
# SVD of this response exceeds 1.
idx = find(max(svd(sys3g))>1)
domain(sys3g(idx(:,1)))
There are several points to observe here. Both svd and max operate on pdms as well as
matrices. Refer to Section 3.2.4 for additional details. The variable idx is a data object
known as an indexlist. In this case the indexlist has three columns: the domain index,
row index and column index. By selecting the domain index the appropriate sub-pdm
can be selected.
The indexlist command creates the indexlist data object from a given three column
matrix. The indexlist data type can also be used for assigning data to part of a pdm.
This is illustrated in the following example.
# Assign the row 2, column 1 element of the 4th domain
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# index of the pdm the value 100.
idxlst = indexlist([4,1,2])
pdm1(idxlst) = 100
Operations on the Independent Variables/Domain
The domain of a pdm is readily changed via the pdm command. The following example
illustrates a common application; changing a frequency domain in Hertz to
radians/second.
# The following scales the domain of a pdm,
# pdm1, by a factor of 2*pi
newpdm = pdm(pdm1,2*pi*domain(pdm1))
Xmath provides a general purpose check which can be used to check whether two pdms
have the same domain. For more information on this function, refer to Section 3.3.1.
The following example illustrates this application.
# Check whether or not pdm1 & pdm2 have the
# same domains.
stat = check(pdm1,pdm2,{samedomain})
# stat = 1 if the domains are the same
3.2.3 Subblocks: selecting input & outputs
Because the Dynamic System is an Xmath data object, standard matrix referencing
can be used to select subsets of the inputs and outputs. This is illustrated in the
following.
# Select inputs 1, 3, 4 & 5 and outputs 2 & 7 from the
# system: bigsys.
subsys = bigsys([1,3:5],[2,7])
The same format has exactly the same function for pdms.
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# Select columns 1, 3, 4 & 5 and rows 2 & 7 from the
# pdm: bigpdm.
subpdm = bigpdm([1,3:5],[2,7])
This referencing format can also be used to assign data to parts of a larger pdm. This is
shown in the following example.
# Replace the 3,2 block of a pdm with its absolute value
pdm1(3,2) = abs(pdm1(3,2))
When selecting subblocks of a Dynamic System or pdm, the appropriate original labels
(input, output and state; row, column and domain) are appended to the subblock.
3.2.4 Basic Functions
In all Xµ functions which operator on Dynamic Systems a matrix input is interpreted
as a system with constant gain: the equivalent of system([],[],[],mat). This
interpretation also applies to binary operations between Dynamic Systems and
matrices in both Xmath and Xµ. For example, +, *, daug,. . . .
In binary operations between pdms and matrices, the matrix is assumed to have a
constant value at each element of the domain of the pdm. Again, this is consistent with
the interpretation of a matrix as a system with no dynamics.
Note that this interpretation works for Xµ and the basic Xmath operations; it will not
necessarily apply to other Xmath modules or more sophisticated Xmath functions.
Augmentation
Augmentation is the building of matrices from component pieces. Dynamic Systems
and pdms can be augmented with the same syntax as matrices. Data objects can be
augmented, side by side, with the command [A, B]. Similarly, [A; B], places A above B.
In Dynamic Systems [A, B] is analogous to summing the outputs of the systems A
and B. [A; B] is analogous to creating a larger system where the input goes to both A
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and B. Augmentation for pdms simply performs the augmentation at each domain
variable. The domains must be the same.
Diagonal augmentation can be performed with the Xµ function daug. This is the
equivalent of the matrix augmentation: [A, 0; 0, B], except that up to 20 arguments
can be augmented in one function call.
Algebraic Operations
In binary operations (e.g. +, -, *) between a Dynamic System and a matrix, the
matrix is assumed to represent the D matrix of a system. In other words, a system with
constant gain (no dynamics).
The transpose operator (’) is well defined for constants, pdms and Dynamic Systems.
In the Dynamic System case it creates the equivalent of the following.
[A,B,C,D] = abcd(sys)
transsys = system(A’,C’,B’,D’)
The conjugate transpose operator (*’) of a Dynamic System creates the adjoint
operator. In other words.
[A,B,C,D] = abcd(sys)
adjsys = system(-A’,-C’,B’,D’)
Random pdms can be created in with the Xµ function randpdm. This is useful for
generating random time domain signals for use in simulations.
Dynamic System Functions
The poles and zeros of Dynamic System can be found with the Xmath core functions
poles and zeros.
Random systems can be created with the Xµ function randsys. The user can also
specify additional system properties; for example, stability, a frequency range for the
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poles, or a zero D term. Generating random systems is useful for simulating systems
with unknown subsystems.
Specialized Xµ functions are provided for useful manipulations of the state. For example
transforming the A matrix to a real valued, 2 × 2 block diagonal form; here referred to
as modal form. These state-space functions are tabulated below.
Description
Xµ
function
state similarity transform simtransform
state reordering orderstate
transform to modal form modalstate
The simtransform function is can also be used on constant matrices and pdms. Xmath
has a transfer function data object which does not have a uniquely defined state.
Applying simtransform or orderstate to a transfer function data object returns a
warning and an unchanged transfer function. Applying modalstate to a transfer
function gives the appropriate state-space representation. With modalstate it is possible
to specify whether the resulting modes are in ascending or descending magnitude order
and whether or not the unstable modes are ordered separately from the stable ones.
pdm Functions
A wide range of matrix functions can also be applied to pdms. Xµ provides several
additional functions which are often of use in typical control applications.
The function spectrad calculates the spectral radius (maximum magnitude eigenvalue)
of a matrix or pdm.
Xmath provides an interpolation function (interpolate) which does only first order
interpolation. There is also a built-in function, spline, for higher order spline fitting.
Zero order interpolation is often required, particularly for looking at fine details in the
inputs of Dynamic System responses. An additional Xµ interpolation function
(interp) is used for this purpose. This command is also useful for mapping data on an
irregularly spaced domain (from experiments for example) to a regular domain.
Simple decimation is easily performed on pdms by direct indexing. The following
example illustrates this.
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# N is the decimation ratio.
smallpdm = bigpdm([1:N:length(bigpdm)])
3.2.5 Continuous to Discrete Transformations
Xmath has a single function, discretize, for calculating continuous to discrete
transformations. Several options are offered including forward and backward difference,
Z-transform equivalent, bilinear and pole-zero matching.
Xmath also has a makeContinuous function which performs the inverse of the
discretize function. Naturally, it is only an inverse if certain pole/zero location and
sampling period relations are assumed to hold.
3.3 Matrix Information, Display and Plotting
3.3.1 Information Functions for Data Objects
The data object information is available in Xmath via the variable window. This
displays the data object classification, row and column information and any associated
comment for each variable in the workspace. For pdms or Dynamic Systems the
domain or state dimension is also displayed. If open, this window is updated
continuously; closing it will speed up calculations.
The Xmath command who displays dimensional information in the command log
window. pdms and Dynamic Systems both appear as three dimensional objects and it
is not possible to distinguish between them using this command alone. The core
function whatis displys the data object type.
Within an Xmath function, data object attributes can be determined with the check or
is functions. These have similar formats, with check being the more powerful. The
function size is used to determine the actual dimensions.
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3.3.2 Formatted Display Functions
It is often useful to consider complex numbers in terms of frequency and damping. This
is particularly applicable when studying the poles or zeros of a system. The command
rifd provides a real-imaginary-frequency-damping formatted display for complex valued
input. The following example illustrates the most common usage.
# display the poles locations of system: sys1
rifd(poles(sys1))
Using the keyword {discrete} will give the display in terms of z-plane rather than the
s-plane. If the input to rifd is a Dynamic System then both the poles and zeros are
calculated and displayed.
3.3.3 Plotting Functions
As pdms are a native data object in Xmath, the standard Xmath plot function will
correctly plot a pdm. Multiple pdms over differing domains can be handled with
repeated calls to the plot function by using the graphical object created from the
previous plot call. The Xmath Basics manual describes this feature in detail.
The Xµ function ctrlplot provides more control system specific plotting capabilities.
Keywords allow the user to easily specify Bode, Nyquist, Nichols and other plots.
Because it handles graphic objects in a similar manner to plot, the resulting plots can
be modified by subsequent plot function calls. An example is given below.
sys1 = 1/makepoly([1,1],"s")
sys2 = 2*sys1*10/makepoly([1,1,10],"s")
w1 = logspace(0.01,10,50)’
w2 = sort([w1;[0.35:0.01:0.65]’])
sys1g = freq(sys1,w1)
sys2g = freq(sys2,w2)
# Bode plots
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g1 = ctrlplot(sys1g,{bode});
g1 = ctrlplot(sys2g,g1,{bode});
g1 = plot({keep=g1,title = "Bode plots",...
legend = ["sys1","sys2"]})?
Bode plots
10
1
0.1
sys1
0.01
sys2
0.001
0.0001
1e-05
0.01
0.1
1
10
Frequency
0
-50
-100
-150
-200
-250
-300
0.01
0.1
1
10
Frequency
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# Nyquist plots
g2 = ctrlplot(sys1g,{nyquist});
g2 = ctrlplot(sys2g,g2,{nyquist});
g2 = ctrlplot(-1,g2,{nyquist,marker=1,line=0});
g2 = plot(g2,{projection="orthographic",...
legend=["sys1","sys3","critical point"],...
title="Nyquist plots"})?
Nyquist plots
0.5
0
-0.5
sys1
sys3
critical point
-1
-1.5
-2
-2
-1
0
1
2
Real
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3.4 System Response Functions
3.4.1 Creating Time Domain Signals
The Signal Analysis Module contains several functions which are useful for building time
domain signals: gcos and gsin. Xµ provides gstep for the creation of stair-step signals.
The example below illustrates the generation of a sine wave and a step function.
# A unit step over 10 seconds
time = 0:10:.1
y1 = gstep(time)
# multiple steps over 10 seconds
sdata = [1;-2;4]
tdata = [3;5;7]
y2 = gstep(time,tdata,sdata)
# Freq: 2 Hz, Ampl: 0.5
y3 = 0.5*gsin(time,2)
Because of the native pdm data type, and the ease in which pdms can be augmented,
vector (or even matrix) valued signals are easily created. For example:
# A function depending on t
time = 0:10:.1
y1 = pdm(exp(0.1*time),time) - gsin(time,3/(2*pi))
# A function independent of t
y2 = uniform(time)
# A 2 x 2 pdm
pt = pdm(time,time)
y3 = [pt/max(time),2*Cos(3*pt+0.2));...
2+Sin(2*pt),sqrt(pt)+0.3*random(pt)]
3.4.2 Dynamic System Time Responses
In Xmath time responses can be calculated by simply multiplying a Dynamic System
by a pdm. A zero-order hold equivalent is used with the sampling interval set to the
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spacing in the input signal pdm. This means that the input pdm must be regularly
spaced. Stair-case input functions with relatively few data points will often give
erroneous results. The input signal should be interpolated (with interp) before being
multiplied with the Dynamic System. The function defTimeRange will calculate a
default integration step size and form the required time vector. This can be useful in
creating an input pdm or as an input to interp. Note that the initial state is specified
within the Xmath Dynamic System.
Discrete time Dynamic Systems do not require an additional command as the time
increment is contained within the Dynamic System data object. The time step in the
input pdm must of course match the discrete system sample time.
Xmath also provides several core functions for common time response calculations: step
calculates the unit step response; impulse calculates the impulse response; and initial
calculates the unforced response from initial conditions specified within the Dynamic
System or provided as input arguments. In each of these functions the user may specify
a time to override the default selected by defTimeRange.
Time response calculations are illustrated in the following example. Several of the
functions discussed in the previous sections are also used.
# Create a second order lightly damped system.
sys = 5/makepoly([1,1,5],"s")
# Create an input signal with a step to +1 at
# 1 second and a step to -1 at 5 seconds.
# 10 second signal is created.
A
u = gstep([0:0.05:10],[0;1;5],[0;1;-1])
# Calculate the time response.
y0 = sys*u
# Repeat this calculation with an initial
# condition of [-1;0]. Note that this is
# only meaningful for a specified realization
# and Xmath forces us to choose one. Here we
# choose that returned as the abcd default
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[a,b,c,d] = abcd(sys)
sys = system(a,b,c,d)
y1 = system(sys,{X0=[-1;0]})*u
# Now plot the result
g1 = ctrlplot(u,{line style=2});
g1 = ctrlplot(y0,g1,{line style=1});
g1 = ctrlplot(y1,g1,{line style=4});
g1 = plot(g1,{!grid})?
2
1
0
-1
-2
-3
0
2
4
6
8
10
The native Xmath time domain simulation, using the * operation, has been
supplemented with an Xµ function: trsp. This provides automatic selection of the
integration time and, if necessary, interpolates the input signal appropriately. trsp also
gives the option of linear interpolation of the input vector and a triangle hold equivalent
discretization of the integration. This simulates the response of the system to signals
which are linearly interpolated between samples. The syntax of the function is
illustrated below.
y = trsp(sys,u,tfinal,{ord,intstep})
sys, u, and tfinal are the dynamic system, input signal and final time respectively. ord
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and intstep are the interpolation order and integration step size. The system must be
continuous. The integration order and sample time are prespecified for discrete time
systems making the * operator is suitable for such simulations.
Xµ provides a sampled data simulation function (sdtrsp) based on the interconnection
structure shown in Figure 3.1.
Figure 3.1: Interconnection structure for sampled data simulation
The signals v and y are the continuous outputs of the continuous system, M(s). The
simulation assumes that y is sampled and used as the input to the discrete system,
N(z). The output of N(z), u(s) is assumed to be held with a zero order hold. All signals
are calculated with the sample spacing determined by the continuous time system
integration step size. The output of N(z) can be delayed by a fraction of the discrete
time system sample time. This is useful for simulating systems with a partial period
calculation delay; a situation which frequently occurs in practice.
3.4.3 Frequency Responses
The Xmath function freq is used for frequency response calculation. The input is a
Dynamic System and a vector of frequency points. The frequency points can
alternatively be specified by minimum and maximum variables and one of two other
choices: the total number of points or phase tracking. If the total number of points is
specified these are logarithmically spaced. If phase tracking is specified the points are
selected such that the phase difference between points is less than a prespecified
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maximum. This can be handy when first examining a high order system. An example is
given below.
# Create a single-input, two-output system.
sys = [1/makepoly([1,0.1,1],"s");...
makepoly([1,1],"s")/makepoly([1,10],"s")]
# The automatic point selection is used.
sysg = freq(sys,{Fmin=0.01,Fmax=100,track})
# Now plot the result
ctrlplot(sysg,{bode})?
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100
10
1
0.1
0.01
0.001
0.0001
1e-05
1e-06
0.01
0.1
1
10
100
Frequency
100
50
0
-50
-100
-150
-200
0.01
0.1
1
10
100
Frequency
Note that the default frequency units in Xmath are Hertz. This applies generically to all
functions where the user specifies frequency information, although many functions allow
radians/second to be specified via a keyword.
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Figure 3.2: Generic Redheffer interconnection structure
3.5 System Interconnection
Interconnections of systems are used extensively in the design, analysis and simulation
calculations. The most general form of interconnection, and the one used in forming
closed loop systems, is the Redheffer (or star) product. The Xµ function starp performs
this operation. This Redheffer product operation is illustrated in Figure 3.2. A MIMO
Dynamic System (or matrix) is created from two MIMO Dynamic Systems.
The syntax of the command to form this interconnection is,
bigsys = starp(sys1,sys2,dim1,dim2)
A more powerful, general, interconnection capability is given with the Xµ function
sysic. This can be used to interconnect subsystems (Dynamic Systems or constant
gain matrices) in an arbitrary manner. This is best illustrated by an example. Consider
the interconnection shown in Figure 3.3.
In each case it is assumed that the systems p, c, and wght, exist in the workspace prior
to calling sysic. The final four-input, three-output system is shown in Figure 3.4.
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Figure 3.3: Example interconnection of subsystems
Figure 3.4: Example interconnected system
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Using sysic to form this interconnection can be considered as four distinct operations.
• Specify the individual subsystems.
• Name and dimension the input signals.
• Specify, algebraically in terms of subsystem outputs or input signals, the output of
the interconnected system.
• Specify, algebraically in terms of subsystem outputs or input signals, the input to
each of the subsystems.
• Name the closed loop system.
The Xµ script used to create the closed-loop system variable clp is shown below. A
single-input, two-output plant is considered. sysic is used to form the closed loop system
for a particular controller. Noise and disturbance signals, in addition to a command
reference, are considered as inputs. Tracking error and the control actuation are the
outputs of interest.
# form the subsystems
p1 = [1;1]*(1/makepoly([1,1,40],"s"));
p = daug(1,1/makepoly([1,1],"s"))*p1;
c = [50*makepoly([1,1],"s")/makepoly([1,10],"s"),-20];
wght = 0.01;
Name the subsystems - these must match with the workspace variables.
snames = ["p"; "c"; "wght"]
Name the final system inputs (names are arbitrary) Parenthesis are used to give a
dimension > 1.
inputs = ["ref"; "dist"; "noise(2)"]
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Specify the outputs in terms of the subsystem names and the input names. Note that
individual outputs of MIMO systems and simple arithmetic combinations can be
specified. Parenthesis specify which output of a MIMO system is to be used.
outputs = ["p(1) + wght"; "7.5*c";
"ref - 1.6*p(1) - 1.6*noise(2) - 1.6*wght"]
Now specify the inputs to each of the named subsystem. The order is important. Note
how the input to a multiple-input system (”c”) is specified.
conx = ["c";...
"ref-1.6*p(1)-1.6*noise(2)-1.6*wght; p(2)+noise(1)";...
"dist"]
Now the final system (clp) is formed.
clp = sysic(snames,inputs,outputs,conx,p,c,wght)
size(clp)?
ans (a row vector) =
3
4
4
rifd(clp)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.5627e+00
-4.4373e+00
-3.0000e+00
-3.0000e+00
0.0000e+00
0.0000e+00
-9.4375e+00
9.4375e+00
1.5627e+00
4.4373e+00
9.9028e+00
9.9028e+00
1.0000
1.0000
0.3029
0.3029
Zeros:
real
imaginary
frequency
damping
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H2
H∞
(rad/sec)
ratio
-1.0000e+00
-5.0000e-01
-5.0000e-01
-1.0000e+01
0.0000e+00
6.3048e+00
-6.3048e+00
0.0000e+00
1.0000e+00
6.3246e+00
6.3246e+00
1.0000e+01
1.0000
0.0791
0.0791
1.0000
The order of the names in the systemname variable, must match the order of the rows in
the connections variable and the order of the last arguments in the sysic function call.
3.6 H2 and H∞ Analysis and Synthesis
This section discusses the synthesis functions available in Xµ. A weighted
interconnection structure is set up so that either H2 or H∞ design methods can be
applied. This discussion assumes that the reader is familiar with the theory and
application of these methodologies. Section 2.3 gives a more detailed overview of the
theory and outlines the algorithms used in the calculations. For specific design examples
and further discussion refer to the demos given in Section 4.1.
3.6.1 Controller Synthesis
The generic synthesis interconnection structure is illustrated in Figure 3.5. The
objective is to design K(s) such that the closed loop interconnection is stable and the
resulting transfer function from w to e (denoted by G(s)) satisfies either an H2 or H∞
norm objective.
Xµ provide functions to calculate the controllers minimizing either the H2 or H∞ norm
of G(s); h2syn and hinfsyn respectively.
Recall from Section 2.3 that the minimizing H2 norm controller (calculated by h2syn) is
unique. The format of this function is given below.
k = h2syn(p,nmeas,ncon)
The variable p is the open loop interconnection structure (P(s) in Figure 3.5). This
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Figure 3.5: Interconnection structure for controller synthesis
structure contains more than just the open loop plant. It typically also contains
frequency dependent weighting functions and specifies the structure of the
interconnection between the open loop plant and the controller. The dimensions of y
and u are specified by nmeas and ncon. The underlying Riccati equations can be solved
by either an eigenvalue or Schur decomposition method. A keyword specifies the desired
Riccati solution method. A simple is example is given at the end of this section.
The hinfsyn function calculates a controller, K(s), which makes kG(s)k∞ ≤ γ for a user
specified γ. It is not possible to make γ arbitrarily small; there is a minimum value for γ
(referred to as γopt) and γopt is not known a priori in a design problem. Therefore
hinfsyn can also perform a bisection search for the smallest γ > γopt and use this value
of γ for the control design. Again, Section 2.3 gives the relevant theoretical details.
The syntax of hinfsyn is illustrated below. The final bound on the achieved γ is
returned as gfin.
[k,gfin] = hinfsyn(p,nmeas,ncon,gamma)
If gamma is a scalar, the controller achieving that γ value is calculated, if one exists. If
gamma is a two element vector a bisection search for the smallest γ value is performed.
The function displays various intermediate calculations related to the eigenvalues of the
Hamiltonian and the positivity of the Riccati solutions. As γ approaches γopt the Riccati
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H2
H∞
equation solution procedure often becomes poorly conditioned. Displaying intermediate
calculation results allows the user to fine tune several tolerances if necessary. The
intermediate Hamiltonian and Riccati solution details are displayed as the bisection
proceeds. The bisection stopping tolerance, Riccati solution tolerances and the Riccati
solution method are specified via keywords. Details on the meaning of these tolerances
are given in Section 2.3.5.
We now give a simple example illustrating the use of these functions. An oscillatory
non-minimum phase SISO system is to be controlled in a unity gain negative feedback
configuration. This example is for pedagogical purposes only and does not illustrate the
generality of the approach with respect to MIMO systems and more general control
configurations. The weighting functions have been chosen to be appropriate for an H∞
design. The performance weight, Wperf, is close to 1/s giving good tracking. The
actuator weight, Wact, increases at high frequency, penalizing fast actuator action. A
sensitivity function and step response comparison have been included to illustrate
typical function calls for these procedures. Inappropriate weight choices make this
comparison blantantly unfair to the H2 approach; the user should not drawn any
conclusions about the relative merits of either approach from this example.
# Set up a simple closed loop problem.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Examine the plant frequency response.
omega = logspace(0.001,100,200)
plantg = freq(plant,omega)
g0 = ctrlplot(plantg,{bode});
g0 = plot(g0,{title="Open loop plant"})?
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Open loop plant
100
10
1
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
0
-100
-200
-300
-400
0.001
0.01
0.1
1
10
100
Frequency
The desired closed loop configuration is illustrated in Figure 3.6.
Figure 3.6: Closed loop configuration
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H2
H∞
Figure 3.7: Weighted design interconnection structure: p
In order to set up the design problem, we consider ref as an unknown input and the
tracking error (input to k), and the actuator signal, u, as outputs to be minimized.
These outputs are weighted with the weights Wperf and Wact respectively. The weighted
interconnection structure for design, p, is illustrated in Figure 3.7.
A more realistic problem would also include weighted noise on the measurement signal,
y. We could also weight the ref input and add weighted disturbances to the plant input
or output.
# Create weights
Wperf = 100/makepoly([100,1],"s")
Wact = makepoly([0.5,0.05],"s")/makepoly([0.05,1],"s")
Wperfg = freq(Wperf,omega)
Wactg = freq(Wact,omega)
g00 = ctrlplot(Wperfg,{logmagplot});
g00 = ctrlplot(Wactg,g00,{logmagplot,line style=4});
g00 = plot(g00,{title="Weighting functions",...
legend=["Wperf";"Wact"],!grid})?
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Weighting functions
100
10
1
0.1
Wperf
Wact
0.01
0.001
0.001
0.01
0.1
1
10
100
Frequency
# Form the weighted interconnection structure
sysnames = ["plant";"Wperf";"Wact"]
sysinp = ["ref";"u"]
sysout = ["Wperf"; "Wact"; "ref-plant"]
syscnx = ["u";...
"ref-plant";... # input to Wperf
"u"] # input to Wact
# input to plant
p = sysic(sysnames,sysinp,sysout,syscnx,plant,...
Wperf,Wact)
# Design Hinf controller
nctrls = 1
nmeas = 1
gmax = 25
gmin = 0
Kinf = hinfsyn(p,nmeas,nctrls,[gmax;gmin])
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
25.0000
Y eig
0.0e+00
Hx eig
X eig
nrho xy
0.0000
p/f
25.000
5.2e-01
1.7e-03
1.0e-02
p
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H2
H∞
12.500
5.2e-01
5.2e-01
5.1e-01
5.0e-01
3.9e-01 -9.8e+01
4.8e-01 1.8e-03
1.7e-03
1.7e-03
1.7e-03
1.7e-03
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
p
p
p
p
f
p
6.250
3.125
1.562
0.781
1.172
Gamma value achieved:
rifd(Kinf)
1.1719
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0000e-02
-1.0347e+00
-2.6846e+00
-2.6846e+00
-1.2692e+01
0.0000e+00
1.0000e-02
1.0347e+00
3.5362e+00
3.5362e+00
1.2692e+01
1.0000
1.0000
0.7592
0.7592
1.0000
0.0000e+00
2.3017e+00
-2.3017e+00
0.0000e+00
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-5.0000e+00
-2.0000e+01
3.1225e-01
-3.1225e-01
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
1.0000
# Design H2 controller
K2 = h2syn(p,nmeas,nctrls)
rifd(K2)
Poles:
real
imaginary
frequency
damping
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(rad/sec)
ratio
-1.4046e-01
-1.4046e-01
-1.5863e+00
-1.5863e+00
-5.2060e+00
-2.3161e-01
2.3161e-01
3.4754e+00
-3.4754e+00
0.0000e+00
2.7087e-01
2.7087e-01
3.8203e+00
3.8203e+00
5.2060e+00
0.5186
0.5186
0.4152
0.4152
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-5.0000e+00
-2.0000e+01
-3.1225e-01
3.1225e-01
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
1.0000
# Look at frequency responses of the controllers
# Kinf and K2
Kinfg = freq(Kinf,omega)
K2g = freq(K2,omega)
g1 = ctrlplot(Kinfg,{logmagplot});
g1 = ctrlplot(K2g,g1,{logmagplot,line style=4});
g1 = plot(g1,{title="Controllers",legend=["Kinf";"K2"]})?
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H2
H∞
Controllers
10
1
0.1
0.01
Kinf
K2
0.001
0.0001
0.001
0.01
0.1
1
10
100
Frequency
# Examine sensitivity functions
sensinf = inv(1 + plant*Kinf)
sensinfg = freq(sensinf,omega)
sens2 = inv(1 + plant*K2)
sens2g = freq(sens2,omega)
g2 = ctrlplot(sensinfg,{logmagplot});
g2 = ctrlplot(sens2g,g2,{logmagplot,line style=4});
g2 = plot(g2,{title="Sensitivity functions",...
legend=["Kinf","K2"],!grid})?
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Sensitivity functions
10
1
Kinf
K2
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
Note that with the interconnection structure shown in Figure 3.7, the closed loop
transfer function from ref to e(1) is simply Wperf*sensinf. If the H∞ of the resulting
closed loop system was less than one, this would guarantee that sensinf was less than
inv(Wperf) at every frequency. Note that from the above plot, this is not quite
satisfied; as we might expect because the closed loop H∞ norm is approximately 1.17.
Now look at some step responses. Use sysic to create an unweighted interconnection
(because the weightings are only for design purposes and are not actually implemented
in the system). We create an open-loop interconnection and close the loop with starp for
each controller.
ic = sysic("plant",["ref";"ctrl"],["plant";"ref-plant"],...
"ctrl",plant)
clpinf = starp(ic,Kinf)
clp2 = starp(ic,K2)
# Examine step response
step = gstep([0:0.1:10],0,1)
yinf = clpinf*step
y2 = clp2*step
g3 = ctrlplot(yinf);
g3 = ctrlplot(y2,g3,{line style=4});
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H2
H∞
g3 = ctrlplot(step,g3,{line style=2});
g3 = plot(g3,{title="Step responses",...
legend=["Kinf";"K2";"input"]})?
Step responses
1.2
1
0.8
0.6
Kinf
K2
input
0.4
0.2
0
0
2
4
6
8
10
3.6.2 System Norm Calculations
Functions are provided for calculating the H2 and H∞ norms of Dynamic Systems. In
the H2 case, this involves the solution of a Lyapunov equation. A bisection method,
involving the calculation of eigenvalues of a scaled Hamiltonian matrix, is required for
the H∞ norm calculation.
The Xµ function for the two norm calculation is called h2norm. The syntax and
operation are self explanatory.
The calculation of the H∞ norm involves the iterative solution of a Riccati equation.
The technique is a generalization of the the theoretical result given in Lemma 2 in
Section 2.3.3. As a result, a tolerance can be specified, and the calculation gives upper
and lower bounds. The function is hinfnorm. The syntax is illustrated below.
[out,omega] = hinfnorm(system,tol)
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Bounds on the H∞ norm are returned as out. An estimate of the frequency where the
norm is achieved is returned as omega. Further control of the iteration is available via
keywords.
The following example calculates the H2 and H∞ norms of each of the closed loop
systems arising from the previous example. Notice that G2 has the minimum
H2 norm and Ginf has the minimum H∞ norm. We can also see that the Ginf has a
slightly lower norm than the bound guaranteed from the hinfsyn function call.
# Calculate the norms of each closed loop system.
Ginf = starp(p,Kinf)
G2 = starp(p,K2)
hinfnorm(Ginf)?
ans (a column vector) =
1.17032
1.16915
hinfnorm(G2)?
ans (a column vector) =
36.2138
36.1776
h2norm(Ginf)?
ans (a scalar) = 2.86197
h2norm(G2)?
ans (a scalar) = 2.78655
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3.7 Structured Singular Value (µ) Analysis and
Synthesis
This section covers the functions used in the D-K iteration procedure. The primary
functions are the calculation of the controller (already discussed), the calculation of µ
and the fitting of rational D scales. Several subroutines used for D scale fitting are
useful in their own regard and are also discussed here. The discussion below assumes
that the reader if familiar with the definition and use of µ; only the implementation
issues are covered here. A simple application example is given here. To get a better idea
of the standard approaches to µ and the D-K iteration, the reader should refer to the
demos in Chapter 4. To find out more about the theoretical aspects of µ, including its
application to robust stability and robust performance problems, refer to Section 2.4.
3.7.1 Calculation of µ
The function mu calculates the structured singular value. µ is is defined as a function of
a matrix and a specified block structure. The most common use involves calculating µ at
each frequency of a system frequency response. For this reason the µ function accepts
pdm input arguments as well as matrices. The block structure is specified to the
function in a coded form.
In general only upper and lower bounds for µ are calculated. The upper bound (for the
square ∆ block case) is given by,
ꢇ
ꢈ
µ(M) ≤ inf σmax DMD−1
,
D∈D
where D is the set of all matrices which commute with the perturbation, ∆. Recall that
the lower bound is given by,
max ρ(MQ) ≤ µ(M),
Q∈Q
where ρ denotes the spectral radius and Q is the set of all unitary perturbations of the
specified block structure. The matrix Q achieving this maximization is a destabilizing ∆.
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The outputs of the µ function are: the upper and lower bounds for µ; the D matrix for
the upper bound; the Q matrix for the lower bounds; and a sensitivity estimate for the
part of the D matrix corresponding to each block in ∆. The sensitivity estimate is
essentially the gradient of the upper bound value with respect to the value of the D
scale. It is useful in weighting the D scale fitting procedure.
The function syntax is shown below.
[mubnds,D,Dinv,Delta,sens] = mu(M,blk)
The upper and lower bounds are returned in mubnds. The variable sens gives a measure
of the sensitivity of the upper bound to the D matrix. Both D and Q are returned as
the matrices (or pdms) D and Delta. Note that the inverse of D, used in the calculation
of the upper bound, is also returned (as Dinv). This is done as, in the non-square ∆
block case, the dimensions of D and D−1 are different.
The block structure is a vector of dimension: number of blocks × 2. For each block the
output and input dimension is specified. To specify a scalar × identity block, the input
dimension is set to zero.
A power iteration, with several random restarts, is used for the lower bound. The upper
bound calculation uses an Osborne balancing method and enhances this with the Perron
vector method for problems with less than 10 blocks. These methods have been found to
be appropriate for the vast majority of practically motivated problems.
New algorithms for these calculations are currently under development. The most
significant enhancement is the ability to calculate µ with respect to structures which
include real valued blocks. Because of the development effort in this direction, a wide
range of calculation options were not provided for the mu function.
The following example gives the simplest matrix with a gap between µ and the D-scale
upper bound. It also illustrates the use of the mu function for constant matrices.
The following is the classic example showing that mu is not equal to its upper bound for
more than three full blocks. We include a random scaling here to give a non-trivial
D-scale.
gamma = 3 + sqrt(3); beta = sqrt(3) -1
a = sqrt(2/gamma); b = 1/sqrt(gamma)
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c = 1/sqrt(gamma); d = -sqrt(beta/gamma)
f = (1+jay)*sqrt(1/(gamma*beta))
psi1 = -pi/2; psi2 = pi
U = [a,0; b,b; c,jay*c; d,f]
V = [0,a; b,-b; c,-jay*c; f*exp(jay*psi1), d*exp(jay*psi2)]
scl = diagonal(random(4,1)+0.1*ones(4,1))
M = scl*U*V*’*inv(scl)
Consider four 1×1 blocks. In this example mu is approximately 0.87.
blk1 = [1,1; 1,1; 1,1; 1,1]
[mubnds1,D1,Dinv1,Delta1] = mu(M,blk1)
max(svd(M))?
# a very crude upper bound
ans (a scalar) = 3.17155
max(svd(D1*M*Dinv1))? # the D scale upper bound
ans (a scalar) = 1
mubnds1?
mubnds1 (a column vector) =
1
0.864113
Consider one 4×4 block (equivalent to maximum singular value).
blk2 = [4,4]
[mubnds2,D2,Dinv2,Delta2] = mu(M,blk2)
Note that the perturbation is such that det(I − MDelta) = 0.
max(svd(D2*M*Dinv2))?
ans (a scalar) = 3.17155
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mubnds2?
mubnds2 (a column vector) =
3.17155
3.17155
det(eye(4,4) - M*Delta2)?
ans (a scalar) = -2.62055e-16 + 5.82345e-17 j
3.7.2 The D-K Iteration
Recall from Section 2.5 that the D-K iteration is used as an approximation to µ
synthesis. This section discusses how Xµ implements this procedure.
The D-K iteration procedure is as follows. The weighted design interconnection
structure is referred to as P. The successive controllers are K i, i = 1,. . . and the
successive closed loop systems are G i, i = 1,. . . . The block structure is coded within
blk; nmeas is the number of controller measurements, and nctrls is the number of
controller actuators outputs.
1. Set i = 1.
2. Design an initial H∞ controller, K 1, for the interconnection structure, P.
K 1 = hinfsyn(P, nmeas, nctrls, gamma limits).
3. Form the closed loop,
G i = starp(P, K i).
4. Calculate µ(G i) as follows.
[bnds, D i, Dinv i, Delta i, sens i] = mu(G i, blk).
This calculation gives the D-scales for the upper bound: D i. Figure 3.8 illustrates
this step.
5. Compare the closed loop to the design specifications; this will involve more than
just the calculation of µ. The user has several options at this point:
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(a) Controller and closed loop are satisfactory so stop the iteration.
(b) The iteration has converged and the controller and closed loop are not
satisfactory. In this case the weighted design problem must be reformulated.
(c) The iteration has not yet converged. Continue with step 6.
6. Fit rational approximations to D i and Dinv i. The function to do this, musynfit,
is described in more detail in Section 3.7.3. A typical function invocation is,
[Dsys i, Dinvsys i] = musynfit(D i, blk, nmeas, nctrls, sens i).
7. Apply rational approximations to D-scales to the weighted interconnection
structure. This is equivalent to,
P D = Dsys i ∗ P ∗ Dinvsys i.
8. Set i = i + 1
9. Design an H∞ controller, K i, for the interconnection structure, P D.
K i = hinfsyn(P D, nmeas, nctrls, gamma limits).
This step is illustrated in Figure 3.9.
10. Go to step 3.
Figure 3.8: µ calculation in the D-K iteration: Step 3 in the enumerated procedure. Note
that µ(G i) ≤ σmax(D i*G i*Dinv i)
The above iteration uses a standard H∞ design. It is possible to use the D-K iteration
procedure with any MIMO design procedure (H2 for example).
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Figure 3.9: H∞ controller design. Step 9 in the enumerated procedure
There is actually another possibility at step 5; numerical problems cause the iteration to
diverge. As γ approaches its optimal value, the numerical properties of the calculation
deteriorate. This may lead to mu(G i) increasing as i is increased. This problem is
observed more often in systems with very lightly damped modes.
A comparison of Figures 3.8 and 3.9 will show that Dsys i is not quite a rational
approximation to D i. The reason is that Dsys i has as inputs, the lower outputs of P.
These are actually passed through an identity for the design of the next controller: K j
(with j = i+1). In other words,
ꢀ
ꢁ
D i
0
0
I
Dsys i ≈
.
This identity is of dimension nmeas × nmeas and is the reason that nmeas and nctrls
must be passed to the musynfit function. Do not confuse this identity with that
corresponding to the last block in D i.
3.7.3 Fitting D Scales
The Xµ D-scale fitting function is musynfit; the syntax is as follows.
[Dsys,Dinvsys] = musynfit(Dmagdata,blk,nmeas,nctrls,weight)
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Both the D and D−1 systems (Dsys and Dinvsys) are returned. The D scale
(Dmagdata) comes from a µ calculation on a closed loop system. However, Dsys and
Dinvsys are required to multiply the open loop system. They must therefore contain the
identity matrices for the inputs and outputs which correspond to the measurements and
controls. This information is not contained in blk and must be specified in the argument
list: nmeas is the number of measurements and ncntrl is the number of controls. The
user can specify a frequency domain weight for the fitting. The variable sens, returned
from the mu function, is a good option.
There are several choices of rational fitting functions available within musynfit. The
D-scale input variables are magnitude data. Phase data, corresponding to a
minimum-phase system is supplied with the Xµ function mkphase. The user has a choice
of functions for the transfer function fitting: the Xmath function tfid and the Xµ
function fitsys. The fitsys function is discussed in more detail in Section 3.7.3 below.
For further information on tfid see the Xmath Basics Manual.
There are several choices of graphical display available to help the user select the most
appropriate fit. These are:
1. The D-scale magnitude data and the last two transfer function fits are displayed.
2. The D-scale magnitude data and the last two transfer function fits are displayed.
An additional plot shows the weighting function.
3. The D-scale magnitude data and the last two transfer function fits are displayed.
An additional plot compares the µ upper bound (which uses the D-scale
magnitude data) to the bound which would be obtained from a frequency response
of the D-scale transfer function fit.
In a problem with n perturbation blocks, there are n − 1 D scales requiring fitting. This
is because one can be chosen as the identity without loss of generality. The D scale for
each block requires user interaction for the selection of the system order. The function
fitsys is called to fit each D scale block. This is available to the user and is described
in more detail below.
Transfer Function Fitting Functions
The underlying functions used for the fitting of each block of the D scale are described
here as they may be of independent interest to the user. Other possible uses include
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creating weights from data and simple system identification.
Xµ provides two user callable functions for fitting SISO transfer functions to data. The
first is mkphase calculates the phase corresponding to a minimum-phase stable system
from magnitude data. This uses the complex cepstrum method described by Oppenheim
and Schafer [80, p. 501] to generate the desired frequency response. The syntax of the
function is given below.
cdata = mkphase(magdata)
The complex cepstrum method is used to generate the complex valued frequency
response,cdata, of a SISO minimum-phase system with magnitude response magdata.
The Xµ function fitsys fits a dynamic system to frequency domain data. The syntax is
illustrated below.
sys = fitsys(data,npoles,nzeros,weight)
The number of poles and zeros in the system, sys, can be independently specified by the
arguments npoles and nzeros. When called from musynfit these are always identical
as an invertible system is required. This may not be appropriate for other applications
of fitsys. For logscale frequency data a weight of 1/s is strongly recommended as this
tends to balance the effects of high and low frequencies in the fit. In musynfit a 1/s
weight is automatically applied and multiplies any other user specified weight.
Chebyshev polynomials are used as basis functions for both the numerator and
denominator polynomials. For further information see the work by Adcock [81]. The
fitsys function is similar to the Xmath function tfid. The algorithm in fitsys has
been fine-tuned for D-scale fitting and usually outperforms tfid in this application.
We will now look at a simple design example using D-K iteration. The plant is a double
integrator with an output perturbation. For more physically meaningful examples refer
to Chapter 4.
# The nominal plant is a double integrator.
# A multiplicative perturbation weight reflects
# increased uncertainty at high frequencies
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plant = 1/makepoly([1,0,-0.01],"s")
Wm = makepoly([1,20],"s")/makepoly([1,200],"s")
omega = logspace(0.01,100,25)
plantg = freq(plant,omega)
Wmg = freq(Wm,omega)
g1 = ctrlplot(plantg,{logmagplot});
g1 = ctrlplot(Wmg,g1,{logmagplot,line style=4});
g1 = plot(g1,{!grid,legend=["Open loop plant";...
"multiplicative perturbation weight"]})?
100
10
1
0.1
0.01
0.001
0.0001
Open loop plant
multiplicative perturbation weight
1e-05
1e-06
0.01
0.1
1
10
100
Frequency
The open-loop system under consideration is illustrated in Figure 3.10.
Figure 3.10: Open loop perturbation model
# Now include some weights for performance:
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Wperf = makepoly([0.01,1],"s")/makepoly([1,0.01],"s")
Wact = 0.1* makepoly([1,1],"s")/makepoly([0.05,1],"s")
Wnoise = 0.01
Wref = makepoly([0.005,1],"s")/makepoly([0.05,1],"s")
Wperfg = freq(Wperf,omega)
Wactg = freq(Wact,omega)
Wnoiseg = conpdm(Wnoise,omega)
Wrefg = freq(Wref,omega)
g2 = ctrlplot(Wperfg,{logmagplot});
g2 = ctrlplot(Wactg,g2,{logmagplot,line style=3});
g2 = ctrlplot(Wnoiseg,g2,{logmagplot,line style=4});
g2 = ctrlplot(Wrefg,g2,{logmagplot,line style=5});
g2 = plot(g2,{title="Performance Weights",y min=0.001,...
legend=["tracking error";"actuator";...
"noise";"reference input"],!grid})?
Performance Weights
100
tracking error
actuator
noise
reference input
10
1
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
Set up a weighted interconnection structure for unity gain negative feedback. This
includes the perturbation weight as well as those for the design performance objectives.
A scalar multiplier is factored out of the perturbation. This is done to give a more
interesting D-scale problem for this example.
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nms = ["plant";"Wm";"Wperf";"Wact";"Wnoise";"Wref"]
inp = ["delt";"ref";"noise";"u"]
outp = ["100*Wm"; "Wperf"; "Wact"; "Wref-0.01*delt-plant-Wnoise"]
cnx = ["u"; "plant"; "Wref-0.01*delt-plant"; "u"; "noise"; "ref"]
p = sysic(nms,inp,outp,cnx,plant,Wm,Wperf,Wact,Wnoise,Wref)
The interconnection structure, p, is illustrated in Figure 3.11.
Figure 3.11: Weighted interconnection structure, p
# Perform an H infinity design.
nmeas = 1
ncntrls = 1
gmin = 0
# number of measurements
# number of controls
gmax = 100
Kinf = hinfsyn(p,nmeas,ncntrls,[gmin;gmax])
G = starp(p,Kinf)
Gg = freq(G,omega)
G11g = Gg(1,1)
# weighted closed loop system
G22g = Gg(2:3,2:3)
rs = max(svd(G11g))
np = max(svd(G22g))
# robust stability
# nominal performance
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blk = [1,1; 2,2]
[rpbnds1,D1,Dinv1,Delta1,sens1] = mu(Gg,blk)
g3 = ctrlplot(np,{log});
# plot on a log-linear scale
g3 = ctrlplot(rs,g3,{log,line style=3});
g3 = ctrlplot(rpbnds1,g3,{log,line style=[4,5]});
g3 = plot(g3,{!grid,title="mu analysis",y lab="Magnitude",...
legend=["nominal perf.";"robust stab.";...
"robust perf.(upper)";"robust perf. (lower)"]})?
mu analysis
4
nominal perf.
robust stab.
robust perf.(upper)
3
robust perf. (lower)
2
1
0
0.01
0.1
1
10
100
# Fit transfer functions to D1 & Dinv1 for a mu
# synthesis iteration
[Ds,Dinvs] = musynfit(D1,blk,nmeas,ncntrls,sens1,{!plotweight})
The following illustrates the musynfit screen display after selecting a 2nd order fit and
then a 4th order fit.
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D scale fit, block: 1
100
Magnitude data
Previous fit, order: 2
New fit, order: 4
10
1
0.1
0.01
0.01
0.1
1
10
100
Frequency (Hz)
# Apply the D scales to another H infinity design
Kmu = hinfsyn(Ds*p*Dinvs,nmeas,ncntrls,[0;10])
# Close the loop around the weighted interconnection
# structure.
Gmu = starp(p,Kmu) # weighted closed loop (2nd it.)
omega = logspace(0.01,100,40)
Gmug = freq(Gmu,omega)
blk = [1,1; 2,2]
[rpbnds2,D2,Dinv2,Delta2,sens2] = mu(Gmug,blk)
# compare mu to the value from the previous iteration.
g4 = ctrlplot(rpbnds2(1,1),{log});
g4 = ctrlplot(rpbnds1(1,1),g4,{log,line style=4});
g4 = plot(g4,{title="mu analysis of robust performance",...
legend=["Kmu";"Kinf"],y lab="Magnitude",!grid})?
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mu analysis of robust performance
4
3.5
3
Kmu
Kinf
2.5
2
1.5
1
0.5
0
0.01
0.1
1
10
100
3.7.4 Constructing Rational Perturbations
For simulation purposes it is useful to be able to construct a rational approximation to
the ∆ returned by the µ calculation. The approach is to choose a ∆ at a particular
frequency, for example the one where µ is at a maximum, and obtain a MIMO system
which has a frequency response (gain and phase) equal to ∆ at that frequency.
The function for this purpose is function mkpert. The syntax is given below.
pertsys = mkpert(Delta,blk,mubnds)
This function takes as arguments the variables Delta, blk, and mubnds. The meaning of
these is identical to the mu case. The frequency selected for the interpolation is that
where the lower bound (in mubnds) is maximum. Alternatively the user can use
keywords to specify a frequency at which to do the interpolation and specify the norm of
the resulting pertsys. pertsys will be an all-pass system.
Monte-Carlo simulation approaches require the ability to generate random perturbations
having the correct block structure. The function for this purpose is randpert and its
usage is illustrated below.
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pert = randpert(blk,{sys,sfreq,complex,pnorm})
The user can specify whether the perturbation is a dynamic system or matrix, and
whether it is real or complex valued, in addition to specifying the norm.
3.7.5 Block Structured Norm Calculations
It is possible to get an idea of the input/output combinations which are limiting the
robust stability or robust performance of a system by examining the elements of the
product, DMD−1 at the critical frequency. Large values indicate a potential problem in
the corresponding input/output pair. For systems with MIMO blocks this can be
simplified to looking at the block norm of the matrix. This is done by partitioning up
the matrix into blocks which correspond to the inputs and outputs of each block of ∆.
Each partition is then replaced by the maximum singular value of the partition. In the
case where all the ∆ blocks are 1 × 1 this reduces to the absolute value of the matrix. If
there is only one ∆ block this is equivalent to calculating the maximum singular value of
the matrix.
The Xµ function for this purpose is called blknorm and its syntax is illustrated below.
normM = blknorm(M,blk,p,{Frobenius})
The user has the option of selecting other p norms (1 ≤ p ≤ ∞) or the Frobenius norm.
The Euclidean norm is the most commonly used and is the default.
3.8 Model Reduction
The model reduction functions described here are often useful in obtaining a lower order
realization of a controller. µ-synthesis controllers for complicated systems are often of
high order. The controller order can often be significantly reduced with very little
degradation in the closed loop performance.
The user should be careful in reducing the order of an open-loop interconnection
structure before doing a design. Very small changes in the interconnection structure can
produce dramatic changes in the resulting closed loop system.
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A greater range of model reduction functions is available in the Model Reduction
Module. Some of the functions described here are cross-licensed with that module.
Section 2.6 describes the theory behind these functions.
3.8.1 Truncation and Residualization
Truncation is provided by the truncate function (cross-licensed with the Model
Reduction Module). Residualization is performed with the Xµ function: sresidualize.
The user must specify the original system and the number of states to be retained. In
both cases the states in the upper left corner of the A matrix are retained.
The following example illustrates the use of these functions and gives an idea of their
error properties.
# Create a five state system for reduction.
a = daug(-0.891334,[-1.20857,0.799042;-0.799042,-1.20857],...
-4.74685,-21.3013)
b = [0.0262569;-0.189601;-0.113729;0.211465;-0.538239]
c = [0.120725,-0.336942,0.397198,-0.700524,-1.02235]
d = 0
sys1 = system(a,b,c,d)
# Reduce to a 3 state system by residualization
# and truncation.
sysout1 = sresidualize(sys1,3)
sysout2 = truncate(sys1,3)
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
sysout1g = freq(sysout1,fHz)
sysout2g = freq(sysout2,fHz)
residerror = sys1g - sysout1g
truncerror = sys1g - sysout2g
g1 = ctrlplot(sys1g,{logmagplot});
g1 = ctrlplot(sysout1g,g1,{logmagplot,line style=2});
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g1 = ctrlplot(sysout2g,g1,{logmagplot,line style=4});
g1 = ctrlplot(residerror,g1,{logmagplot,line style=5});
g1 = ctrlplot(truncerror,g1,{logmagplot,line style=6});
g1 = plot(g1,{!grid,legend=["original system";...
"residualized system";"truncated system";...
"residualization error";"truncation error"]})?
0.1
0.01
0.001
original system
residualized system
truncated system
residualization error
0.0001
truncation error
1e-05
0.01
0.1
1
10
100
Frequency
3.8.2 Balanced Realizations
The function balmoore (cross-licensed from the Model Reduction Module) produces a
balanced realization and optionally truncates it. The can be used to obtain a balanced
realization and the Hankel singular values without necessarily truncating the system.
The following illustrates the use of the function using the same system as in the
truncate and sresidualize example.
# Balanced truncation.
[sysout3,hsv] = balmoore(sys1,{nsr=3})
sysout3g = freq(sysout3,fHz)
balerr = sys1g - sysout3g
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# Displaying the Hankel singular values shows which
# states are close to unobservable and uncontrollable.
hsv?
hsv (a column vector) =
0.0741834
0.0726887
0.0264105
0.000146401
2.7699e-07
# Compare to the errors from the previous example.
g2 = ctrlplot(sys1g,{logmagplot});
g2 = ctrlplot(sysout3g,g2,{logmagplot,line style=2});
g2 = ctrlplot(balerr,g2,{logmagplot,line style=4});
g2 = ctrlplot(residerror,g2,{logmagplot,line style=5});
g2 = ctrlplot(truncerror,g2,{logmagplot,line style=6});
g2 = plot(g2,{!grid,legend=["original system";...
"balanced truncation";"error: balmoore";...
"error: sresidualize";"error: truncate"]})?
0.1
0.01
0.001
0.0001
original system
balanced truncation
error: balmoore
error: sresidualize
error: truncate
1e-05
1e-06
0.01
0.1
1
10
100
Frequency
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3.8.3 Hankel Singular Value Approximation
The function ophank (also cross-licensed from the Model Reduction Module) is used to
perform optimal Hankel norm approximation. Recall from Section 2.6.3 that there is an
astable system achieving the lower bound. The unstable part of this system is returned
as the second argument.
The following applies the Hankel norm approximation technique to the previous
example. The unstable part of the optimal approximation is sysout4u and the astable
system, sysout4 + sysout4u would achieve the lower bound for the approximation
error. The Hankel singular values can also be obtained by this function.
# Optimal Hankel singular value approximation
[sysout4,sysout4u,hsv] = ophank(sys1,{nsr=3})
sysout4g = freq(sysout4,fHz)
hankerr = sys1g - sysout4g
# Compare to the errors from the previous example.
g3 = ctrlplot(sys1g,{logmagplot});
g3 = ctrlplot(sysout4g,g3,{logmagplot,line style=2});
g3 = ctrlplot(hankerr,g3,{logmagplot,line style=6});
g3 = ctrlplot(balerr,g3,{logmagplot,line style=4});
g3 = ctrlplot(residerror,g3,{logmagplot,line style=5});
g3 = plot(g3,{!grid,legend=["original system";...
"Hankel norm approximation";"error: ophank";...
"error: balmoore";"error: sresidualize"]})?
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0.1
0.01
0.001
0.0001
1e-05
1e-06
original system
Hankel norm approximation
error: ophank
error: balmoore
error: sresidualize
0.01
0.1
1
10
100
Frequency
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Chapter 4
Demonstration Examples
4.1 The Himat Example
The following demo can be run by executing the following Xmath command:
execute file = "$XMATH/demos/xMu/himatdemo"
4.1.1 Problem Description
The Himat is a small scale remotely piloted aircraft built to investigate high
maneuverability fighter aircraft design. The vehicle was flight tested in the late 1970s.
The example studied here considers control of only the longitudinal dynamics. These are
taken to be uncoupled from the lateral-direction dynamics. The nominal model and
control objectives are described by Safonov et al. [82]. Further details can be found in
the work by Hartman et al. [83] and Merkel et al. [84]. The vehicle is currently attached
to the outside wall of the Museum of Science in Los Angeles.
A four state rigid body model describes the dynamics. The states can be assigned the
following physical meanings,
127
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δv Perturbations along the velocity vector.
α Angle of attack. I.e. angle between the velocity vector and the aircraft’s longitudinal
axis.
q Rate-of-change of aircraft attitude angle.
θ Aircraft attitude angle.
Control can be exerted via the elevon and canard, denoted by δe and δc respectively.
The angle of attack (α) and attitude angle (θ) are available as direct measurements.
A weighted output disturbance rejection problem will be considered. This problem also
encompasses other maneuvering objectives. The closed-loop perturbation model of the
vehicle is illustrated in Figure 4.1.
Figure 4.1: Himat open-loop perturbation model
Note that Wp, Wdel, Himat and K are each two-input, two-output systems. Similarly,
the inputs dist and pertin are two element vector signals. The desired result is the
four-input, four-output system, denoted by clp, and shown in Figure 4.2. This is used in
the design example given below.
4.1.2 State-space Model of Himat
The state space description of the Himat plane is given below. The states of the plant
model are: forward speed (v), angle-of-attack (α), pitch rate (q) and pitch angle (θ).
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Figure 4.2: Interconnection structure for the himat design example
The inputs are the elevon position and the canard position. The outputs that are to be
kept small are angle-of-attack (α) and pitch angle (θ).
The commands required to enter the state-space description are simply matrix
assignments for each of A, B, C and D. Note that the states, inputs and outputs are
named.
a = [-0.0226, -36.6000, -18.9000, -32.1000;...
0, -1.9000,
0.0123, -11.7000, -2.6300,
0.9830,
0;...
0;...
0]
0,
0,
1.0000,
b = [
0,
0;...
0;...
-0.4140,
-77.8000, 22.4000;...
0,
0]
c = [0, 57.3000, 0,
0;...
0,
0, 0, 57.3000]
d = zeros(2,2)
himat = system(a,b,c,d)
comment himat "Himat vehicle state-space model"
states = ["forward speed";"angle-of-attack";"pitch rate";"pitch
angle"]
inputs = ["elevon";"canard"]
outputs = ["angle-of-attack";"pitch angle"]
himat = system(himat,{stateNames=states,inputNames=inputs,...
outputNames=outputs})?
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himat (a state space system) =
A
-0.0226
0
0.0123
0
-36.6
-1.9
-11.7
0
-18.9
0.983
-2.63
1
-32.1
0
0
0
B
0
0
0
-0.414
-77.8
0
22.4
0
C
0
0
57.3
0
0
0
0
57.3
D
0
0
0
0
X0
0
0
0
0
State Names
-----------
forward speed
angle-of-attack
pitch rate
pitch angle
Input Names
-----------
elevon
canard
Output Names
------------
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angle-of-attack
pitch angle
System is continuous
4.1.3 Creating a Weighted Interconnection Structure for Design
The multiplicative input perturbation weight, Wdel, is constructed as a transfer function.
The weight is,
50(s + 100)
(s + 10000)
Wdel
=
.
The output error weight, Wp, is created as,
0.5(s + 3)
(s + 0.03)
Wp =
.
This will be used as the performance weight. The appropriate commands are:
wdel = makepoly([50,5000])/makepoly([1,10000])
wp = makepoly([0.5,1.5])/makepoly([1,0.03])
These can be displayed on a frequency response plot for comparison purposes.
om1 = logspace(0.001,10000,100)
wdelg = freq(wdel,om1)
# frequency vector (Hz)
comment wdelg "frequency response of wdel"
wpg = freq(wp,om1)
comment wpg "frequency response of wp"
gph1 = ctrlplot([wdelg,wpg],{logmagplot});
gph1 = plot(gph1,{title="Weights for HIMAT",!grid,...
legend=["Perturbation weight";"Performance weight"]})?
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Weights for HIMAT
100
10
1
Perturbation weight
Performance weight
0.1
0.001
0.01
0.1
1
10
Frequency
100
1000
10000
The perturbation weight, Wdel, should actually be two-input, two-output. This is also
true of the performance weight, Wp. In this example, we are weighting each performance
channel identically. There is no requirement to do this and if we were more concerned
with errors in α than errors in θ, then the appropriate channel would have a large weight
applied. The function daug is used to make the 2×2 systems.
wdel = daug(wdel,wdel)
comment wdel "perturbation weight"
wp = daug(wp,wp)
comment wp "performance weight"
The design interconnection structure is now created with the sysic function. The result
is a state space system (8 states, 6 inputs and 6 outputs) called himat ic.
The first two inputs and outputs correspond to the multiplicative perturbation block.
Inputs and outputs 3 and 4 are the disturbance inputs & error outputs. The
measurements going to the controller appear on outputs 5 & 6. The control actuation is
put into the system on inputs 5 & 6.
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sysn = ["himat";"wdel";"wp"]
in = ["pert(2)";"dist(2)";"control(2)"]
out = ["wdel";"wp";"himat + dist"]
inter = ["control + pert"; "control"; "himat + dist"]
himat ic = sysic(sysn,in,out,inter,himat,wdel,wp)
comment himat ic "Himat design interconnection structure"
4.1.4
H
Design
∞
The next step is to design an H control law for Himat. The function hinfsyn designs
∞
an H control law based on the interconnection structure provided. hinfsyn requires
∞
the design interconnection structure, number of measurements, number of controls and
single gamma or bisection bounds on γ.
Optional input arguments are the tolerance for terminating the γ iteration (tol), an
epsilon for Hamiltonian jω-axis eigenvalues (epr) and the epsilon for the Riccati
solution positive definite tests (epp). Two Riccati solution methods are provided:
eigenvalue or Schur (default) decomposition. hinfsyn returns the control law, k and the
gamma value achieved, gf1.
In this example, the system interconnection structure is himat ic, with 2 measurements,
2 controls, a γ lower bisection bound of 0.8, a γ upper bisection bound of 6, a tolerance
on the γ iteration of 0.05, and we’ll use the eigenvalue decomposition method to solve
the Riccati equations. The default values of epr (0.5*sqrt(eps)) and epp (1e-6) will be
used for the epsilon tests.
gamma bounds = [0.8;6.0]
nmeas = 2
nctrls = 2
# 2 measurements: attack angle & pitch
# 2 controls: elevon & canard
comment nmeas "number of controller measurements"
comment nctrls "number of controller outputs"
[k1,gf1] = hinfsyn(himat ic,nmeas,nctrls,gamma bounds,{tol=0.05})
Test bounds:
0.8000
<
gamma <=
6.0000
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gamma
6.000
3.400
2.100
1.450
1.775
1.613
1.694
1.653
Hx eig
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
X eig
5.6e-05
5.7e-05
5.9e-05
6.4e-05
6.1e-05
6.2e-05
6.1e-05
6.2e-05
Hy eig
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
2.3e-02
Y eig
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
nrho xy
0.0626
0.2020
0.5798
1.4678
0.8652
1.1028
0.9725
1.0343
p/f
p
p
p
f
p
f
p
f
Gamma value achieved:
1.6938
comment k1 "controller: iteration 1"
comment gf1 "gamma value: iteration 1"
g1 = starp(himat ic,k1)
comment g1 "closed loop: iteration 1"
An H control law has been designed which achieves an infinity norm of 1.6938 for the
∞
interconnection structure provided. First, we will examine aspects of the controller that
was just designed, starting with the controller poles.
rifd(k1)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.2609e-02
-3.0000e-02
-3.0000e-02
-1.3833e+01
-9.8959e+01
-1.4712e+02
-1.4712e+02
-7.4256e+03
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
9.6846e+01
-9.6846e+01
0.0000e+00
2.2609e-02
3.0000e-02
3.0000e-02
1.3833e+01
9.8959e+01
1.7613e+02
1.7613e+02
7.4256e+03
1.0000
1.0000
1.0000
1.0000
1.0000
0.8353
0.8353
1.0000
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Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.2516e-02
-1.7226e+00
-3.0272e+00
-3.1034e+01
-1.0000e+04
-1.0000e+04
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
2.2516e-02
1.7226e+00
3.0272e+00
3.1034e+01
1.0000e+04
1.0000e+04
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
Next, a magnitude plot of the frequency response of k1 is plotted to check that it looks
reasonable.
om2 = logspace(0.5,5000,25)
k1 g = freq(k1,om2)
comment k1 g "frequency response of k1"
comment om2 "frequency vector (Hz)"
gph2 = ctrlplot(k1 g,{bode});
gph2 = plot(gph2,{title="Controller: k1"})?
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Controller: k1
1
0.1
0.01
0.001
0.1
1
10
100
1000
10000
Frequency
200
100
0
-100
-200
-300
0.1
1
10
100
1000
10000
Frequency
Onto the closed loop, first checking that it is stable by looking at the pole positions.
rifd(g1)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
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-2.2517e-02
-2.2600e-02
-3.0000e-02
-3.0000e-02
-2.9369e+00
-2.9974e+00
-4.8310e+00
-6.5876e+00
-5.8350e+01
-5.8350e+01
-8.8792e+01
-8.8792e+01
-9.9778e+01
-7.4258e+03
-1.0000e+04
-1.0000e+04
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
-5.6049e+01
5.6049e+01
-4.2881e+01
4.2881e+01
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
2.2517e-02
2.2600e-02
3.0000e-02
3.0000e-02
2.9369e+00
2.9974e+00
4.8310e+00
6.5876e+00
8.0909e+01
8.0909e+01
9.8604e+01
9.8604e+01
9.9778e+01
7.4258e+03
1.0000e+04
1.0000e+04
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.7212
0.7212
0.9005
0.9005
1.0000
1.0000
1.0000
1.0000
Zeros:
ans (a scalar) =
0
Now calculate a closed loop frequency response. This results in a 4x4 pdm: g1g
g1g=freq(g1,om2)
comment g1g "frequency response of g1"
The singular values of the closed loop system are calculated. The maximum of the
singular values over frequency is also calculated and compared to γ. As γ is a
guaranteed upper bound on the infinity norm, this value should be less.
g1gs = svd(g1g)
comment g1gs "closed loop singular value: iteration 1"
gph3 = ctrlplot(g1gs,{log});
gph3 = plot(gph3,{!grid,...
title="Singular value plot of the closed loop"})?
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Singular value plot of the closed loop
2
1.5
1
0.5
0
0.1
1
10
100
1000
10000
4.1.5 µ Analysis of the H Controller
∞
The H control law can be analyzed using µ-analysis. The closed-loop system, g1, has 4
∞
inputs and 4 outputs. The first two inputs and outputs correspond to the uncertainty
block, and the second two correspond to the disturbance rejection block, or performance
block. Therefore, we can define the uncertainty inputs and outputs as a full 2×2
uncertainty block and the disturbance rejection inputs and outputs as a full 2×2
performance block. If the µ value for this control design is 1, then we are able to achieve
robust performance for the set of defined weights and this control design.
The mu function analyzes the robust performance and stability of the closed loop system.
The syntax of the function is:
[bnds,D,Dinv,Delta,sens] = mu(M,blk)
The variable M is usually the frequency response of the closed loop system. blk defines
the structure of the perturbations. In this example the block structure is two 2×2
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complex valued blocks.
The upper and lower bounds of the µ function are returned in bnds. Also returned are
the scaling matrices, D and Dinv, corresponding to the upper bound. The smallest
destabilizing perturbation at each frequency is returned as Delta. The variable sens is
the sensitivity of the upper bound to the D and Dinv scaling matrices. This will be
useful as a weighting function for fitting transfer functions to the D and Dinv scaling
blk = [2,2;2,2]
[bnds1,D1,D1inv,Delta1,sens1] = mu(g1g,blk)
comment blk "perturbation block structure"
comment bnds1 "mu bounds: iteration 1"
comment D1 "D scale: iteration 1"
comment D1inv "D inverse scale: iteration 1"
comment Delta1 "worst case perturbation: iteration 1"
comment sens1 "D scale sensitivity: iteration 1"
We plot the maximum singular value and µ on the same plot. The performance and
stability specifications have been achieved if µ is less than one at all frequencies.
gph4 = ctrlplot(g1gs(1,1),{log,line style=4});
gph4 = ctrlplot(bnds1,gph4,{log,line style=[1,3]});
gph4 = plot(gph4,{title="Max. singular value and mu comparison",...
legend=["max. singular value";"mu upper bound";...
"mu lower bound"],!grid})?
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Max. singular value and mu comparison
1.8
1.6
1.4
1.2
1
max. singular value
0.8
0.6
0.4
mu upper bound
mu lower bound
0.1
1
10
100
1000
10000
Note that µ(g1g) is not less than one at all frequencies — we have not met the design
objectives. A D-K iteration will be used to lower µ and improve the robust performance
with respect to these objectives.
4.1.6 Fitting D-scales for the D-K Iteration
In some cases the H controller is adequate for our purposes. It is not necessarily the
∞
controller which gives the best robust performance for our system — it essentially
ignores the structure in the perturbations.
The D-K iteration procedure using the D and Dinv scaling matrices in the µ calculation
to set up an H problem which will usually give better robust performance.
∞
The first step is to pre and post multiply the interconnection, (himat ic in this case)
with D and Dinv. Two things need to be done first. The D and Dinv matrices produced
by µ are pdms and the interconnection is a state-space system. We must first fit transfer
functions to the D and Dinv magnitude data before we can do the multiplication.
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The second thing to note is that the interconnection structure has the additional control
inputs and measurement outputs. The D and Dinv systems must be augmented with
identities corresponding to these additional inputs and outputs.
The musynfit function performs both of these operations. The syntax of musynfit is:
[Dsys,Dinvsys] = musynfit(D,blk,nmeas,nctrls,sens,oldDsys)
The variables D and sens come directly from the mu function. The block structure is
specified by blk and nmeas and nctrls are the number of measurements and controls
respectively.
The outputs are the dynamic systems which approximate D and Dinv. The new scaled
H
problem can be set up with
∞
new ic = Dsys * old ic * Dinvsys.
D-K iteration involves iterating between calculating and fitting D-scales and designing
controllers, K.
If there are N blocks in the µ problem set up, then the D scale matrices have N − 1
different transfer functions that require fitting. The Nth transfer function is taken to be
unity.
We recommend choosing a 3rd order transfer function for the fit. This increases the
number of states in the interconnection structure by 3*(size of block)*2. A different
order can be chosen — which will lead to a slightly different controller in the subsequent
analysis.
Note that g1g is also passed to musynfit. This will provide the user with a comparison
between the calculated upperbound and that based on the rational fit. This comparison
is useful in deciding between fits of differing orders.
[D1sys,D1invsys] = musynfit(D1,blk,nmeas,nctrls,sens1,g1g,{Hertz})
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D scale fit, block: 1
10
1
Magnitude data
Previous fit, order: 0
New fit, order: 3
0.1
0.01
0.1
1
1
1
10
100
1000
10000
10000
10000
Frequency (Hz)
Upper bound comparison, block: 1
10
Data based bound
Previous fit bound, order: 0
New fit bound, order: 3
1
0.1
0.1
10
100
1000
Frequency (Hz)
D scale fit weight for block: 1
10
1
0.1
0.01
0.1
10
100
1000
Frequency (Hz)
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comment D1sys "system approx. to D1"
comment D1invsys "system approx. to D1inv"
4.1.7 Design Iteration #2
The new D scales can be pre and post multiplied onto the orginal interconnection
structure.
himat ic2 = D1sys * himat ic * D1invsys
[,,nx] = size(himat ic2)
display "himat ic2 now has " + string(nx) + " states"
himat ic2 now has 20 states
comment himat ic2 "interconnection for iteration 2"
Note the increase in states due to the inclusion of the D scales. A new H controller
∞
can now be designed.
gamma bounds = [0.9,6.0]
[k2,g2,gf2] = hinfsyn(himat ic2,nmeas,nctrls,...
gamma bounds,{tol=0.05})
Test bounds:
gamma
0.9000
Hx eig X eig
2.3e-02 -1.1e-09
2.3e-02 -3.1e-12
2.3e-02 -1.2e-08
<
gamma <=
Hy eig
1.7000
Y eig
nrho xy
0.3120
0.6390
1.1028
0.8180
0.9423
1.0171
p/f
p
p
f
p
1.700
1.300
1.100
1.200
1.150
1.125
2.2e-02 -2.5e-27
2.2e-02 -1.4e-29
2.2e-02 -3.4e-22
2.2e-02 -2.1e-15
2.2e-02 -1.1e-15
2.2e-02 -5.8e-31
2.3e-02
2.2e-14
2.3e-02 -7.7e-10
2.3e-02 -1.9e-08
p
f
Gamma value achieved:
1.1500
comment k2 "controller: iteration 2"
comment gf2 "gamma value: iteration 2"
Note that we calculate the new closed loop using the original interconnection: himat ic.
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g2 = starp(himat ic,k2)
comment g2 "closed loop: iteration 2"
[,,nx] = size(k2)
display "k2 now also has " + string(nx) + " states"
k2 now also has 20 states
This design probably resulted in a control law which achieved an infinity norm of
approximately 1.1 for the new interconnection structure.
k2 has been designed to reduce µ(g2). The singular values may actually get worse. We
will see that this is the case here.
The stability of g2 is examined with the function check. As used here, it will return a 1
if it is stable and a zero if not. A frequency response is then calculated.
check(g2,{stable})
ans (a scalar) = 1
g2g=freq(g2,om2)
comment g2g "g2 frequency response"
The µ analysis is repeated to assess the closed loop robust performance of the µ based
controller (k2). The command format is identical to the last time.
[bnds2,D2,D2inv,Delta2,sens2] = mu(g2g,blk)
gph6 = ctrlplot(bnds2,{log,line style=[1,4]});
gph6 = ctrlplot(bnds1,gph6,{log,line style=[3,5]});
gph6 = plot(gph6,{title="Robustness analysis for the g2 system",...
legend=["g2: mu upper bound";"g2: mu lower bound";...
"g1: mu upper bound";"g1: mu lower bound"],!grid})?
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Robustness analysis for the g2 system
1.8
1.6
1.4
1.2
1
g2: mu upper bound
g2: mu lower bound
g1: mu upper bound
g1: mu lower bound
0.8
0.6
0.4
0.1
1
10
100
1000
10000
comment bnds2 "mu bounds: iteration 2"
comment D2 "D scale: iteration 2"
comment D2inv "D inverse scale: iteration 2"
comment Delta2 "worst case perturbation: iteration 2"
comment sens2 "D scale sensitivity: iteration 2"
Here we have only done a single D-K iteration. We are now much closer to the
specifications although we still do not quite meet them. In practice several more
iterations could be run to further improve the µ design. The next iteration would use
the commands:
[D2sys,D2invsys] = musynfit(D2,blk,nmeas,nctrls,sens2,g2g)
himat ic3 = D2sys*himat ic*D2invsys
k3 = hinfsyn(himat ic3,,nmeas,nctrls,gamma bounds,tol=0.05)
and to analyze it we would use the commands:
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g3 = starp(himat ic,k3)
g3g = freq(g3,omega)
[bnds3,D3,D3inv,Delta3,sens3] = mu(g3g,blk)
At this point we could do another iteration (to get k4) or perhaps run some simulations
to check out k3 more thoroughly.
4.1.8 Simulation Comparison with a Loopshaping Controller
A loopshaping design is performed and compared to the H and µ designs. As the
∞
loopshaping design procedure is relatively standard, only the final controller is given
here (klp).
a =[-5.8928e-02,-6.3295e+00,-1.0440e+00,...
-1.6190e-02, 1.8469e+00, 1.0405e-02;...
-1.0283e+00,-1.5776e+03,-1.1900e+03,...
-9.9924e+00, 1.7895e+01, 3.0329e+00;...
1.0440e+00, 1.0146e+03,-4.2005e+01,...
-1.0953e+00, 2.1755e+02, 9.9809e-01;...
1.3287e-02, 9.6897e+00,-5.9979e-01,...
-2.5448e-02, 1.3634e+01, 4.5286e-02;...
6.4800e+00, 4.0284e+01,-5.9317e+02,...
-1.5215e+01,-2.3726e+04,-8.1459e+01;...
1.3057e-02,-4.0389e+00,-1.2179e+00,...
-1.4332e-02,-5.3180e+01,-2.3590e-01]
b =[1.2425e+00, -6.0707e-02; ...
1.5602e+01, -6.9151e+01; ...
-1.0002e+01, 5.5826e-01; ...
-1.2752e-01, 1.1238e-01; ...
-6.9355e+01, -1.4693e+01; ...
-1.4459e-01 ,-1.2232e-01]
c = [3.5741e-01,7.9890e-01,2.8247e+00,...
8.7133e-02,-7.0893e+01,-1.7902e-01;...
1.1915e+00,7.0885e+01,9.6107e+00,...
1.4594e-01, 3.3669e-01,-6.1799e-02]
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d = zeros(2,2)
klp = system(a,b,c,d)
comment klp "loop shape controller"
We will compare the designs, with no error or uncertainty weights, for the nominal case
and with a perturbation block of ∆ = [0.1,0;0,-0.1] for the input multiplicative
perturbation.
The time response will be from 0 to 2 seconds with a sample time of 0.01 seconds. We’ll
look at a unit step input into the first channel.
The unweighted closed loop system is now formed for each controller. The inputs and
outputs to the perturbations are closed around a ∆ = 0 perturbation initially. This is
equivalent to selecting the nominal inputs and outputs.
sysnames = ["himat";"wdel"]
invars = ["pert(2)"; "dist(2)"; "control(2)"]
outvars = ["wdel"; "himat+dist"; "himat+dist"]
connections = ["control + pert";"control"]
gsim = sysic(sysnames,invars,outvars,connections,himat,wdel)
comment gsim "unweighted interconnection"
gsim mu = starp(gsim,k2)
gsim hinf = starp(gsim,k1)
gsim lp = starp(gsim,klp)
comment gsim mu "closed loop system: mu ctrl"
comment gsim hinf "closed loop system: hinf ctrl"
comment gsim lp "closed loop system: klp ctrl"
The nominal systems are studied first.
gsim mu nom = gsim mu(3:4,3:4)
gsim hinf nom = gsim hinf(3:4,3:4)
gsim lp nom = gsim lp(3:4,3:4)
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comment gsim mu nom "nominal closed loop sys: mu ctrl"
comment gsim hinf nom "nominal closed loop sys: hinf ctrl"
comment gsim lp nom "nominal closed loop sys: klp ctrl"
A step disturbance is introduced into the first channel.
time = 0:2:0.01
u = gstep(time)
u = [u;0*u]
# a unit step is the default
# put zero into the other channel
comment u "simulation input"
y mu nom = gsim mu nom*u
y hinf nom = gsim hinf nom*u
y lp nom = gsim lp nom*u
comment y mu nom "nominal response: mu ctrl"
comment y hinf nom "nominal response: hinf ctrl"
comment y lp nom "nominal response: klp ctrl"
All controllers perform well on the nominal system. This can be seen by plotting the
time histories of each.
gph7 = plot(y mu nom,{rows=3,row=1,grid,...
title="Kmu step dist. response (nominal)",...
x lab="time",y max=1.5,y min=-1});
gph7 = plot(y hinf nom,gph7,{row=2,grid,...
title="Kinf step dist. response (nominal)",...
x lab="time",y max=1.5,y min=-1});
gph7 = plot(y lp nom,gph7,{row=3,grid,...
title="Klp step dist. response: (nominal)",...
x lab="time",y max=1.5,y min=-1})?
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Kmu step dist. response (nominal)
1.5
1
0.5
0
-0.5
-1
0
0
0
0.5
0.5
0.5
1
1.5
1.5
1.5
2
2
2
time
Kinf step dist. response (nominal)
1.5
1
0.5
0
-0.5
-1
1
time
Klp step dist. response: (nominal)
1.5
1
0.5
0
-0.5
-1
1
time
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The loopshaping design gives a decoupled response. Both the H∞ and µ designs trade
decoupling for speed of response and, as we shall see, robustness with respect to
perturbations.
The simulation is repeated with a perturbation of size 0.1. Note that this is only 10% of
the size perturbation that we were analyzing and designing for in the above.
delta = [.1,0;0,-0.1]
comment delta "example perturbation"
gsim mu pert = starp(delta,gsim mu)
gsim hinf pert = starp(delta,gsim hinf)
gsim lp pert = starp(delta,gsim lp)
comment gsim mu pert "perturbed closed loop sys: mu ctrl"
comment gsim hinf pert "perturbed closed loop sys: hinf ctrl"
comment gsim lp pert "perturbed closed loop sys: klp ctrl"
y mu pert = gsim mu pert*u
y hinf pert = gsim hinf pert*u
y lp pert = gsim lp pert*u
comment y mu pert "perturbed response: mu ctrl"
comment y hinf pert "perturbed response: hinf ctrl"
comment y lp pert "perturbed response: klp ctrl"
The loop shaping controller performs poorly on the perturbed system. Again the time
response of each system is plotted.
gph8 = plot(y mu pert,{rows=3,row=1,grid,...
title="Kmu step dist. response (perturbed)",...
x lab="time",y max=1.5,y min=-1});
gph8 = plot(y hinf pert,gph8,{row=2,grid,...
title="Kinf step dist. response (perturbed)",...
x lab="time",y max=1.5,y min=-1});
gph8 = plot(y lp pert,gph8,{row=3,grid,...
title="Klp step dist. response: (perturbed)",...
x lab="time",y max=1.5,y min=-1})?
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Kmu step dist. response (perturbed)
1.5
1
0.5
0
-0.5
-1
0
0
0
0.5
0.5
0.5
1
1.5
1.5
1.5
2
2
2
time
Kinf step dist. response (perturbed)
1.5
1
0.5
0
-0.5
-1
1
time
Klp step dist. response: (perturbed)
1.5
1
0.5
0
-0.5
-1
1
time
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The loopshaping controller had good nominal performance and very poor robust
performance. This was illustrated with a relatively small perturbation. The difference
between the µ and H controllers was small in both the nominal and perturbed cases.
∞
This may not always be the case for several reasons.
Only a single D-K iteration was performed here. Further iterations would further
improve the performance of the µ controller.
The perturbation chosen for the above simulation was not the worst case one. Note that
here, each of the three closed loop systems will have a different worst case perturbation.
To find these, perform a µ calculation on each closed loop system and use mkpert to
construct the appropriate perturbations.
The theoretical measure of performance is the H norm of the closed loop transfer
∞
function. When assessing different controllers by simulation we are applying additional,
unformalized performance measures.
In this case, the performance and perturbation channels were about equally scaled. The
resulting D-scales were within an order of magnitude of unity. Choosing a different set
of units for α and/or θ would change the H norm of the result without changing µ. A
∞
poor choice of engineering units could therefore lead to a larger difference between the
H
controller and the µ controller.
∞
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4.2 A Simple Flexible Structure Example
The demonstration script, jplphBdemo.ms, runs through a D-K iteration design for a
simple flexible structure problem.
The following demo can be run by executing the following Xmath command:
execute file = "$XMATH/demos/xMu/jplphBdemo"
where $XMATH is the path to your Xmath source location.
The problem comes from an experiment in the NASA Control Structures Interaction
(CSI) program and is located at the Jet Propulsion Laboratory. A more complete
description is given by Spanos et al. [85, 86, 87].
The problem involves controlling the length of a laser path which reflects off of a series
of mirrors mounted on a flexible structure. One measurement is available to the
controller: the pathlength. There are two actuators, a voice coil and a piezo-electric,
each driving a mirror in the path. These actuators are effectively in parallel at the same
point. They differ significantly in their characteristics and uncertainty descriptions.
This is a relatively simple problem and the user should be able to achieve similar results
through standard classical techniques. The code given here serves as a suitable template
for user written design scripts. It should be noted that a large number of frequency
points are used in this design — perhaps more than necessary — and this significantly
slows down the calculation of µ.
4.2.1 The Control Design Problem
The optical configuration is illustrated schematically in Figure 4.3. The laser system
consists of a laser and optical interferometer. It is mounted on a fixed optical bench.
The laser is directed to the actuated mirrors mounted on the flexible structure. The
path continues back to the optical bench where it hits a target mirror and reflects back
through the structure mirrors to the interferometer. The interferometer gives a
measurement of the optical pathlength to an accuracy of 2.5 nm.
Vibrations in the flexible structure affect the optical pathlength and the objective is to
maintain a constant pathlength in the presence of such vibrations. Complicating the
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Figure 4.3: Schematic diagram of the JPL Phase B optical testbed design problem
problem is the fact that the voice-coil mirror assembly has significant mass and its
movement excites a mode in the structure. An identically driven counterbalance
effectively makes the piezo-electric actuator reactionless.
The closed loop design problem is illustrated in Figure 4.4.
Figure 4.4: JPL Phase B closed loop optical control configuration
We now construct a model of the voice-coil mirror actuator, vcmodel. This model has
been obtained from identification experiments. Note that it has an oscillatory pole pair
at a frequency very close to an oscillatory zero pair.
vcmodel1 = makepoly([1,.056309,1162.8],"s")/...
makepoly([1,.78756,1189.1],"s")
vcmodel2 = -252861.0/makepoly([1,1.0936,19.673],"s")
vcmodel = vcmodel1*vcmodel2
delete vcmodel1 vcmodel2
size(vcmodel)?
ans (a row vector) =
1
1
4
In the frequency range of interest the piezo mirror driver can nominally be modeled as a
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static gain. We will include some dynamic uncertainty in the actual design.
piezo = 1
4.2.2 Creating the Weighted Design Interconnection Structure
The weighted, open-loop design interconnection structure is illustrated in Figure 4.5.
For clarity, the two perturbations, ∆1 and ∆2, have been shown inside the structure. We
have included perturbation models for both the actuators, involving the weights
Wavoice, Wmvoice and Wmpiezo.
In addition there is an error performance weight, Wperf, and two actuator penalties,
Wactv and Wactp. The relative sizes of the noise and disturbance inputs are specified by
the weights Wnoise and Wdist respectively.
Figure 4.5: Weighted open-loop interconnection structure for the JPL Phase B optical
design problem
The voice coil additive weight, Wavoice, is simply a constant. Because the nominal voice
coil model rolls off at 40db/decade this weight indicates significant gain and phase
uncertainty for the frequencies where |Wavoice| > |vcmodel|.
Wavoice = 1
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The additive weight clearly provides for significant high frequency uncertainty. A
multiplicative weight models the low frequency uncertainty. The value selected is
somewhat arbitrary and can be considered as a tunable design weight.
Wmvoice = 0.1
A multiplicative perturbation is used to model uncertainty in the piezo driver.
Experimental observations suggest a 5% deviation from nominal across the frequency
range of interest.
Wmpiezo = 0.05
Wdist is a weight for disturbances on the flexible structure. The magnitude of this has
been estimated by comparing experimental disturbance responses to the voice coil
system response. This weight can also be considered as a variable in the design problem.
Increasing the weight will place more emphasis on disturbance rejection in the final
design.
Wdist = 0.01
The noise weight, Wnoise is selected based on what is necessary to achieve a final
resolution of 10nm.
Wnoise = 0.005
The performance weight, Wperf is used to specify system performance up to 80 Hz. We
use a 2nd order Butterworth filter to roll off at frequencies beyond 80 Hz.
Wperf = butterworth(2,{Fc=80,dT=0})
There are actuator penalties for the piezo driver, Wactp, and the voice coil driver, Wactv
.
The piezo seems to have usable bandwidth out to about 400 Hz. We use a first order roll
up, starting at 100 Hz. to penalize higher frequency actuation. The piezo penalty is set
at about 200 times greater than the voice coil penalty. This is because the piezo
actuator will saturate at 30 micrometers and the voice coil saturates at 6 mm.
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F = 100
Wactp = makepoly([1/(2*pi*F),1],"s")/...
makepoly([1/(200*pi*F),1],"s")
Wactp = Wactp*4
The lower frequency response of the voice-coil system means that the Wactv weight
should begin rolling up at around 10 Hz.
F = 10
Wactv = makepoly([1/(2*pi*F),1],"s")/...
makepoly([1/(200*pi*F),1],"s")
Wactv = Wactv*0.02
Both of these weights can be adjusted to trade between the relative responses from the
voice-coil and the piezo. This should be done after examining simulations, or
experimental closed loop data.
The weights are concatenated together for easier display.
weights = [Wperf;Wavoice;Wmvoice;Wmpiezo;Wdist;Wnoise;Wactp;Wactv]
We now construct the weighted design interconnection structure using sysic.
ssnames = ["vcmodel"; "Wavoice"; "Wmvoice"; "Wmpiezo";...
"Wdist"; "Wnoise"; "Wperf"; "Wactp"; "Wactv"; "piezo"]
inps = ["d1i"; "d2i"; "dist"; "noise"; "vact"; "pact"]
ops = ["Wavoice";"Wmvoice";"Wmpiezo";"Wperf";"Wactv";"Wactp";...
"d1i + vcmodel + d2i + piezo + Wnoise"]
cnx = ["Wdist + vact"; "Wdist + vact"; "vcmodel"; "piezo";...
"dist"; "noise"; "d1i + vcmodel + d2i + piezo + Wnoise";...
"pact"; "vact"; "pact"]
P = sysic(ssnames,inps,ops,cnx,vcmodel,Wavoice,Wmvoice,Wmpiezo,...
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Wdist,Wnoise,Wperf,Wactp,Wactv,piezo)
size(P)?
ans (a row vector) =
7
6
8
Now select a frequency grid for calculating the frequency responses. Some additional
points are included near the oscillatory modes.
omega = logspace(1,1000,15)’
omega = sort([omega; [5.275:0.05:5.625]’; [4.5:0.25:6.5]’])’
Examine the frequency response of the open-loop system.
vcmodelg = freq(vcmodel,omega)
gph1 = ctrlplot(vcmodelg,bode);
gph1 = plot(gph1,{title="voice coil model",!grid})?
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voice coil model
100000
10000
1000
100
10
1
0.1
0.01
0.001
1
10
100
1000
Frequency
100
80
60
40
20
0
-20
-40
1
10
100
1000
Frequency
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And examine the design weights.
weightsg = freq(weights,omega)
gph2 = ctrlplot(weightsg,{logmagplot});
gph2 = plot(gph2,legend=["Wperf";"Wavoice";"Wmvoice";...
"Wmpiezo";"Wdist";"Wnoise";"Wactp";"Wactv"],...
title="Design weights",!grid)?
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Design weights
100
Wperf
Wavoice
Wmvoice
Wmpiezo
Wdist
Wnoise
Wactp
10
Wactv
1
0.1
0.01
0.001
1
10
100
1000
Frequency
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4.2.3 Design of an H∞ Controller
An H∞ design is now performed. Recall that we have one interferometer measurement
and two controller outputs.
nmeas = 1
ncon = 2
glimits = [0;20]
[Khinf,gamma] = hinfsyn(P,nmeas,ncon,glimits,{tol=0.25})
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
20.0000
Y eig
Hx eig
X eig
nrho xy
0.0007
0.0026
0.0109
0.0487
0.4059
p/f
20.000
10.000
5.000
2.500
1.250
2.8e-02
2.8e-02
2.8e-02
2.8e-02
2.8e-02
3.1e-07
3.1e-07
3.1e-07
3.1e-07
3.1e-07
2.8e-01 -6.5e-17
2.8e-01 -4.5e-17
2.8e-01 -1.7e-16
2.8e-01
2.8e-01 -1.6e-16
p
p
p
p
p
0.0e+00
0.625
0.938
1.094
1.172
2.5e-07 *******
2.8e-02 -1.9e+06
2.8e-01 -2.2e-16 *******
f
2.8e-01 -1.6e-16
2.8e-01 -1.9e-16
1.7198
1.0727
0.5933
f
f
p
2.8e-02
2.8e-02
3.1e-07
3.1e-07
2.8e-01
0.0e+00
Gamma value achieved:
1.1719
Now form the closed loop and check the pole positions. As expected, it is stable.
Ghinf = starp(P,Khinf)
rifd(Ghinf)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.8248e-02
-2.8248e-02
-2.7911e-01
-2.7911e-01
-2.9727e+01
-2.9727e+01
3.4100e+01
-3.4100e+01
3.4212e+01
-3.4212e+01
-3.0010e+01
3.0010e+01
3.4100e+01
3.4100e+01
3.4213e+01
3.4213e+01
4.2241e+01
4.2241e+01
0.0008
0.0008
0.0082
0.0082
0.7038
0.7038
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-3.5543e+02
-3.5543e+02
-6.4598e+02
-6.4598e+02
-3.8358e+02
-3.8358e+02
-1.6934e+03
-2.8107e+03
-6.2832e+03
-6.2832e+04
-3.5543e+02
3.5543e+02
-3.3537e+01
3.3537e+01
-5.3777e+02
5.3777e+02
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
5.0265e+02
5.0265e+02
6.4685e+02
6.4685e+02
6.6055e+02
6.6055e+02
1.6934e+03
2.8107e+03
6.2832e+03
6.2832e+04
0.7071
0.7071
0.9987
0.9987
0.5807
0.5807
1.0000
1.0000
1.0000
1.0000
Zeros:
We can also look at the controller poles. It turns out that our controller is stable.
rifd(Khinf)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.8002e-02
-2.8002e-02
-3.5543e+02
-3.5543e+02
-6.4828e+02
-1.1873e+03
-1.1873e+03
-2.8870e+03
3.4100e+01
-3.4100e+01
-3.5543e+02
3.5543e+02
0.0000e+00
-3.0188e+02
3.0188e+02
0.0000e+00
3.4100e+01
3.4100e+01
5.0265e+02
5.0265e+02
6.4828e+02
1.2251e+03
1.2251e+03
2.8870e+03
0.0008
0.0008
0.7071
0.7071
1.0000
0.9692
0.9692
1.0000
Zeros:
We also look at a frequency response of the controller. Recall that it is single-input,
two-output.
Khinfg = freq(Khinf,omega);
gph3 = ctrlplot(Khinfg,{bode});
gph3 = plot(gph3,{title="Controller: Khinf"})?
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Controller: Khinf
1
0.1
0.01
0.001
0.0001
1
10
100
1000
Frequency
200
150
100
50
0
-50
-100
1
10
100
1000
Frequency
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4.2.4 Robustness Analysis
The block structure has two perturbation ∆ blocks and a “performance” block. The two
voice-coil perturbations are put into a single 1×2 block as they enter the system at the
same point. Note that this is not identical to two separate blocks — for example a
perturbation in which both blocks have magnitude one is now longer included. The
advantage of doing this is that it give one less block in the resulting analysis and design
problem. This makes the µ calculation easier (we now have three blocks and so the
upper bound is actually equal to µ) and gives one less D-scale to be approximated in the
D-K iteration.
blk = [1,2; 1,1; 2,3]
The frequency response of the closed loop system is calculated. The nominal
performance test simply involves checking the H∞ norm of the nominal closed loop
system. This is a maximum singular value test.
Ghinfg = freq(Ghinf,omega);
npbnds = norm(svd(Ghinfg(4:6,3:4)),inf)
Robust stability is a µ test as there are two pertubation blocks. Note that here we
calculate µ with respect to the G11 partition and use a block structure containing only
the perturbation blocks.
[rsbnds,Drs,Drsinv,Deltars,sensrs] = mu(Ghinfg(1:3,1:2),[1,2; 1,1])
Robust performance is a µ test on the entire G matrix.
[rpbnds,D,Dinv,Delta,sens] = mu(Ghinfg,blk)
These results are plotted. Note that µ tests give bounds and in this case the upper and
lower bounds are almost indistinguishable.
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gph4 = ctrlplot(npbnds,{log,line style=4});
gph4 = ctrlplot(rsbnds,gph4,{log,line style=[3,5]});
gph4 = ctrlplot(rpbnds,gph4,{log,line style=[1,2]});
gph4 = plot(gph4,{!grid,legend=["Nom perf";"Rob stab (up bnd)";...
"Rob stab (lw bnd)"; "Rob perf (up bnd)";...
"Rob perf (lw bnd)"],title="mu analysis of Ghinf"})?
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perturbed closed loop: vc actuator
0.015
0.01
Khinf
Kmu
0.005
0
-0.005
-0.01
-0.015
0
0.5
1
1.5
2
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4.2.5 D-K Iteration
We will now perform one D-K iteration to generate the controller Kmu. Significant
robustness and performance improvement is achieved with only one iteration.
Transfer functions are fit to the D-scales from the previous robust performance µ test.
Here we preselect an order of 2 for each D-scale. This has been found to give a
satisfactory result.
[Dsys,Dinvsys] = musynfit(D,blk,nmeas,ncon,sens,[],2)
Now a new weighted interconnection is formed by pre- and post-multiplying by the
D-scale approximations. A second H∞ design is performed to get Kmu.
Pd = Dsys*P*Dinvsys
glimits = [0;20]
Kmu = hinfsyn(Pd,nmeas,ncon,glimits,{tol=0.1,epr=1e-10,epp=1e-4})
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
20.0000
Y eig
Hx eig
X eig
nrho xy
0.0001
0.0006
0.0022
0.0090
0.0364
0.1518
0.7425
p/f
20.000
10.000
5.000
2.500
1.250
0.625
0.312
0.156
0.234
2.9e-02
2.9e-02
2.9e-02
2.9e-02
2.9e-02
2.8e-02
2.8e-02
3.5e-10
3.3e-10
3.7e-10
3.1e-10
3.6e-10
3.4e-10
3.8e-10
1.2e-01 -1.2e-15
1.2e-01 -3.8e-16
1.2e-01 -3.7e-16
1.2e-01 -3.7e-16
1.2e-01 -1.8e-15
1.2e-01 -2.1e-16
1.2e-01 -2.2e-29
p
p
p
p
p
p
p
f
f
f
p
2.1e-02 -4.3e+06
1.5e-01 -2.2e-17 18.9961
2.6e-02
2.7e-02
2.7e-02
3.6e-10
3.8e-10
3.9e-10
1.3e-01 -2.3e-16
1.2e-01 -9.8e-17
1.2e-01 -1.9e-16
1.7666
1.0735
0.8828
0.273
0.293
Gamma value achieved:
0.2930
Note that this value of γ is significantly lower than even the µ value from the Ghinf
closed loop system. Again, both the controller and closed loop system are stable.
A frequency response of Kmu is calculated and plotted.
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Kmug = freq(Kmu,omega)
gph5 = ctrlplot(Kmug,{bode});
gph5 = plot(gph5,{title="Controller: Kmu"})?
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Controller: Kmu
1
0.1
0.01
0.001
0.0001
1
10
100
1000
Frequency
200
150
100
50
0
-50
-100
1
10
100
1000
Frequency
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We now examine the robustness properties of the new closed loop system. We already
know that the robust performance test will be less than the γ value from the H∞
synthesis above (in this case 0.2930). The results are again displayed graphically.
Gmug = freq(Gmu,omega)
npbnds = norm(svd(Gmug(4:6,3:4)),inf)
[rsbnds,Drs,Drsinv,Deltars,sensrs] = mu(Gmug(1:3,1:2),[1,2; 1,1])
[rpbnds,D,Dinv,Delta,sens] = mu(Gmug,blk)
gph6 = ctrlplot(npbnds,{log,line style=4});
gph6 = ctrlplot(rsbnds,gph6,{log,line style=[3,5]});
gph6 = ctrlplot(rpbnds,gph6,{log,line style=[1,2]});
gph6 = plot(gph6,{!grid,legend=["Nom perf";"Rob stab (up bnd)";...
"Rob stab (lw bnd)"; "Rob perf (up bnd)";...
"Rob perf (lw bnd)"],title="mu analysis of Gmu"})?
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mu analysis of Gmu
0.3
0.25
0.2
0.15
Nom perf
Rob stab (up bnd)
Rob stab (lw bnd)
Rob perf (up bnd)
Rob perf (lw bnd)
0.1
0.05
0
1
10
100
1000
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4.2.6 A Simulation Study
Now the two controllers (Khinf and Kmu) are studied by simulation. An unweighted
interconnection is set up with sysic and starp is used to close the loop for each
controller.
ssnames = ["vcmodel"; "Wavoice"; "Wmvoice"; "Wmpiezo"; "piezo"]
inps = ["d1i"; "d2i"; "dist"; "noise"; "vact"; "pact"]
ops = ["Wavoice"; "Wmvoice"; "Wmpiezo";...
"d1i + vcmodel + d2i + piezo + noise"; "vact"; "pact";...
"d1i + vcmodel + d2i + piezo + noise"]
cnx = ["dist + vact"; "dist + vact"; "vcmodel"; "piezo"; "pact"]
Pnom =
sysic(ssnames,inps,ops,cnx,vcmodel,Wavoice,Wmvoice,Wmpiezo,piezo)
Random inputs are created for the noise and structure disturbances. Both are normally
distributed. These are passed through their respective performance weights to give
signals of the appropriate size (and if necessary frequency content). Both signals are
plotted.
u1 = randpdm(400,1,1,{Dlast=2,regular,zeromean})
u2 = randpdm(400,1,1,{Dlast=2,regular,zeromean})
u = [Wdist*u1; Wnoise*u2]
gph7 = ctrlplot(u(1,1));
gph7 = plot(gph7,{title="Simulation: disturbance"})?
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Simulation: disturbance
0.006
0.004
0.002
0
-0.002
-0.004
-0.006
0
0.5
1
1.5
2
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gph8 = ctrlplot(u(2,1));
gph8 = plot(gph8,{title="Simulation: noise",...
y lab="micrometers"})?
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Simulation: noise
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
0
0.5
1
1.5
2
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A nominal response is calculated by setting ∆ = 0. To get the open-loop simulation
model we close the unweighted interconnection structure with a controller equal to zero.
deltazero = zeros(2,3)
Kzero = zeros(2,1)
nomolp = starp(deltazero,Pnom)
nomolp = starp(nomolp,Kzero)
yolp = nomolp*u
gph9 = ctrlplot(yolp(1,1));
gph9 = plot(gph9,{title="open loop beam length",...
y lab="micrometers"})?
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open loop beam length
10
5
0
-5
-10
0
0.5
1
1.5
2
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Now we consider the closed-loop nominal response with Khinf and Kmu. The closed-loop
system happens to have a large number of high frequency poles which do not contribute
significantly to the response. They have the effect of forcing a very fine time
discretization in the simulation, resulting in a long calculation time. We remove all poles
of frequency greater than 100 rad/sec. by residualization. This is not intended as a
general procedure — in some situations the high frequency behavior will be significant.
nomclp = starp(deltazero,Pnom)
nomclphinf = starp(nomclp,Khinf)
nomclpmu = starp(nomclp,Kmu)
fmax = 100;
nlfpoles = sum(abs(poles(nomclphinf))<fmax)
str = "Residualizing closed loop system to " + ...
string(nlfpoles)+" states"
display(str)
Residualizing closed loop system to 6 states
nsysclphinf = modalstate(nomclphinf)
nsysclphinf = sresidualize(nsysclphinf,nlfpoles)
nlfpoles = sum(abs(poles(nomclpmu))<fmax)
nsysclpmu = modalstate(nomclpmu)
nsysclpmu = sresidualize(nsysclpmu,nlfpoles)
The closed-loop responses are calculated and plotted.
yclphinf = nsysclphinf*u
yclpmu = nsysclpmu*u
gph10 = ctrlplot(yclphinf(1,1));
gph10 = ctrlplot(yclpmu(1,1),gph10);
gph10 = plot(gph10,{legend=["Khinf";"Kmu"],...
title="closed loop beam length",...
y lab="micrometers"})?
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closed loop beam length
0.4
0.3
0.2
0.1
0
Khinf
Kmu
-0.1
-0.2
-0.3
0
0.5
1
1.5
2
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gph11 = ctrlplot(yclphinf(2,1));
gph11 = ctrlplot(yclpmu(2,1),gph11);
gph11 = plot(gph11,{legend=["Khinf";"Kmu"],...
title="closed loop: vc actuator"})?
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closed loop: vc actuator
0.015
0.01
Khinf
Kmu
0.005
0
-0.005
-0.01
-0.015
0
0.5
1
1.5
2
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gph12 = ctrlplot(yclphinf(3,1));
gph12 = ctrlplot(yclpmu(3,1),gph12);
gph12 = plot(gph12,{legend=["Khinf";"Kmu"],...
title="closed loop: piezo actuator"})?
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closed loop: piezo actuator
0.03
0.02
0.01
0
Khinf
Kmu
-0.01
-0.02
0
0.5
1
1.5
2
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Note that Kmu achieves better performance at the expense of greater actuator effort.
We will now repeat this simulation for a perturbed system. A bad ∆ is chosen and
scaled to have norm 0.5. This is obtained from destabilizing ∆ calculated for the µ lower
bound. An all-pass system is fitted to ∆ at one frequency to create a real-rational
perturbation. The frequency selected is that where µ is at its maximum. In this case we
choose the ∆ that comes from the robust stability µ test. This gives the perturbation
which comes closest to destabilizing the closed loop system. Using ∆ from the robust
performance test would allow us to create a perturbation which minimizes the robust
performance of the closed loop system.
Here we use the worst-case ∆ for Kmu. This is almost certainly not the worst-case ∆ for
Khinf. Again, the closed loop systems are residualized prior to calculating the responses.
deltabad = mkpert(Deltars,blk(1:2,1:2),rsbnds,{pnorm=0.5})
badclp = starp(deltabad,Pnom)
badclphinf = starp(badclp,Khinf)
badclpmu = starp(badclp,Kmu)
nlfpoles = sum(abs(poles(badclphinf))<fmax)
str = "Residualizing closed loop system to "+...
string(nlfpoles)+" states"
display(str)
Residualizing closed loop system to 6 states
nbclphinf = modalstate(badclphinf)
nbclphinf = sresidualize(nbclphinf,nlfpoles)
nlfpoles = sum(abs(poles(badclpmu))<fmax)
nbclpmu = modalstate(badclpmu)
nbclpmu = sresidualize(nbclpmu,nlfpoles)
The responses are calculated and plotted.
ybclphinf = nbclphinf*u
ybclpmu = nbclpmu*u
gph13 = ctrlplot(ybclphinf(1,1));
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gph13 = ctrlplot(ybclpmu(1,1),gph13);
gph13 = plot(gph13,{legend=["Khinf";"Kmu"],...
title="perturbed closed loop beam length",...
y lab="micrometers"})?
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Controller: Khinf
1
0.1
0.01
0.001
0.0001
1
10
100
1000
Frequency
200
150
100
50
0
-50
-100
1
10
100
1000
Frequency
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gph14 = ctrlplot(ybclphinf(2,1));
gph14 = ctrlplot(ybclpmu(2,1),gph14);
gph14 = plot(gph14,{legend=["Khinf";"Kmu"],...
title="perturbed closed loop: vc actuator"})?
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perturbed closed loop: vc actuator
0.015
0.01
Khinf
Kmu
0.005
0
-0.005
-0.01
-0.015
0
0.5
1
1.5
2
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gph15 = ctrlplot(ybclphinf(3,1));
gph15 = ctrlplot(ybclpmu(3,1),gph15);
gph15 = plot(gph15,{legend=["Khinf";"Kmu"],...
title="perturbed closed loop: piezo actuator"})?
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perturbed closed loop: piezo actuator
0.03
0.02
0.01
0
Khinf
Kmu
-0.01
-0.02
-0.03
0
0.5
1
1.5
2
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Bibliography
[1] J. C. Doyle, “Lecture notes on advances in multivariable control.” ONR/Honeywell
Workshop, Minneapolis, MN., 1984.
[2] J. Doyle, “Structured uncertainty in control system design,” in Proc. IEEE Control
Decision Conf., pp. 260–265, 1985.
[3] A. K. Packard, What’s new with µ: Structured Uncertainty in Multivariable
Control. PhD thesis, University of California, Berkeley, 1988.
[4] R. S. Smith, Model Validation for Uncertain Systems. PhD thesis, California
Institute of Technology, 1990.
[5] J. Doyle and G. Stein, “Multivariable feedback design: Concepts for a
classical/modern synthesis,” IEEE Trans. Auto. Control, vol. AC-26, pp. 4–16, Feb.
1981.
[6] M. Morari and E. Zafiriou, Robust Process Control. New Jersey: Prentice-Hall,
1989.
[7] The MathWorks, Inc., Natick, MA, µ-Analysis and Synthesis Toolbox (µ-Tools),
1991.
[8] J. Doyle, K. Lenz, and A. K. Packard, “Design examples using µ synthesis: Space
shuttle lateral axis FCS during reentry,” in Proc. IEEE Control Decision Conf.,
pp. 2218–2223, dec 1986.
[9] G. J. Balas, Robust Control of Flexible Structures, Theory and Experiment. PhD
thesis, California Institute of Technology, 1990.
[10] G. J. Balas and J. C. Doyle, “Identification of flexible structures for robust
control,” IEEE Control Sys. Magazine, vol. 10, pp. 51–58, June 1990.
193
Download from Www.Somanuals.com. All Manuals Search And Download.
[11] G. J. Balas and J. C. Doyle, “Robust control of flexible modes in the controller
crossover region,” in Proc. Amer. Control Conf., 1989.
[12] G. J. Balas, A. K. Packard, and J. Harduvel, “Application of µ-synthesis techniques
to momentum management and attitude control of the Space Station,” in AIAA
Guidance, Navigation and Cont. Conf., 1991.
[13] R. S. Smith, C.-C. Chu, and J. L. Fanson, “The design of H controllers for an
∞
experimental non-collocated flexible structure problem,” IEEE Trans. Control Syst.
Tech., 1993. in press.
[14] J. Fanson, C.-C. Chu, B. Lurie, and R. Smith, “Damping and structural control of
the JPL phase 0 testbed structure,” J. Intell. Material Sys. & Struct., vol. 2,
pp. 281–300, July 1991.
[15] R. S. Smith, J. Doyle, M. Morari, and A. Skjellum, “A case study using µ:
Laboratory process control problem,” in Proc. Int. Fed. Auto. Control, vol. 8,
pp. 403–415, 1987.
[16] R. S. Smith and J. Doyle, “The two tank experiment: A benchmark control
problem,” in Proc. Amer. Control Conf., vol. 3, pp. 403–415, 1988.
[17] S. Skogestad, M. Morari, and J. C. Doyle, “Robust control of ill-conditioned plants:
High-purity distillation,” IEEE Trans. Auto. Control, vol. 33, pp. 1092–1105,
December 1988.
[18] S. Skogestad, “Correction to “Robust control of ill-conditioned plants: High purity
distillation”,” IEEE Trans. Auto. Control, vol. 34, p. 672, June 1989.
[19] J. Doyle and A. K. Packard, “Uncertain multivariable systems from a state space
perspective,” in Proc. Amer. Control Conf., vol. 3, pp. 2147–2152, 1987.
[20] A. K. Packard and J. C. Doyle, “The complex structured singular value,”
Automatica, vol. 29, no. 1, pp. 71–109, 1993.
[21] D. R. Hamburg and M. A. Shulman, “A closed loop A/F control model for internal
combustion engines,” Soc. Automotive Eng., no. 800826, 1980.
[22] B. G. Morton and R. M. McAfoos, “A mu-test for robustness analysis of a
real-parameter variation problem,” in Proc. Amer. Control Conf., 1985.
[23] J. Doyle and A. Packard, “Uncertain multivariable systems from a state space
perspective,” in Proc. Amer. Control Conf., pp. 2147–2152, IEEE, 1987.
Download from Www.Somanuals.com. All Manuals Search And Download.
[24] D. L. Laughlin, K. G. Jordan, and M. Morari, “Internal model control and process
uncertainty: mapping uncertainty regions for SISO controller design,” Int. J. of
Control, vol. 44, no. 6, pp. 1675–1698, 1986.
[25] R. S. Smith and M. Dahleh, eds., The Modeling of Uncertainty in Control Systems:
Proceedings of the 1992 Santa Barbara Workshop. 391 pgs., Springer-Verlag, 1994.
[26] M. Gevers, “Connecting identification and robust control: A new challenge,” in
Proc. IFAC Symp. on Identification & System Parameter Estimation, vol. 1,
pp. 1–10, 1991.
[27] A. Helmicki, C. Jacobson, and C. Nett, “H identification of stable lsi systems: A
∞
scheme with direct application to controller design,” Proc. Amer. Control Conf.,
pp. 1428–1434, 1989.
[28] G. Gu and P. P. Khargonekar, “Linear and nonlinear algorithms for identification in
H∞ with error bounds,” in Proc. Amer. Control Conf., pp. 64–69, 1991.
[29] A. J. Helmicki, C. A. Jacobson, and C. N. Nett, “Control oriented system
identification: A worst-case/deterministic approach in H ,” IEEE Trans. Auto.
∞
Control, pp. 1163–1176, 1991.
[30] P. Ma¨kil¨a and J. Partington, “Robust approximation and identification in H∞,”
Proc. Amer. Control Conf., pp. 70–76, 1991.
[31] G. Gu and P. P. Khargonekar, “Linear and nonlinear algorithms for identification in
H∞ with error bounds,” in IEEE Trans. Auto. Control, vol. 37, pp. 953–963, 1992.
[32] G. Gu and P. P. Khargonekar, “A class of algorithms for identification in H∞,” in
Automatica, vol. 28, pp. 299–312, 1992.
[33] G. Gu, P. P. Khargonekar, and Y. Li, “Robust convergence of two-stage nonlinear
algorithms for identification in H∞,” in Syst. and Control Letters, vol. 18,
pp. 253–263, 1992.
[34] R. G. Hakvoort, “Worst-case system identification in H : error bounds and
∞
optimal models,” in Selected Topics in Identification Modelling and Control, Delft
University Press, Vol. 5 1992.
[35] E.-W. Bai, “On-line H2, H and pointwise uncertainty bound quantification in
∞
identification of restricted complexity models,” in Proc. IEEE Control Decision
Conf., pp. 1719–1724, 1992.
[36] G. Goodwin and M. Salgado, “Quantification of uncertainty in estimation using an
embedding principle,” in Proc. Amer. Control Conf., 1989.
Download from Www.Somanuals.com. All Manuals Search And Download.
[37] R. Kosut, M. Lau, and S. Boyd, “Parameter set identification of systems with
uncertain nonparametric dynamics and disturbances,” in Proc. IEEE Control
Decision Conf., vol. 6, pp. 3162–3167, 1990.
[38] G. Goodwin, B. Ninness, and M. Salgado, “Quantification of uncertainty in
estimation,” in Proc. Amer. Control Conf., pp. 2400–2405, 1990.
[39] B. M. Ninness and G. C. Goodwin, “Robust frequency response estimation
accounting for noise and undermodeling,” in Proc. Amer. Control Conf.,
pp. 2847–2851, 1992.
[40] R. S. Smith and J. Doyle, “Model invalidation — a connection between robust
control and identification,” in Proc. Amer. Control Conf., pp. 1435–1440, 1989.
[41] J. M. Krause, “Stability margins with real parameter uncertainty: Test data
implications,” in Proc. Amer. Control Conf., pp. 1441–1445, 1989.
[42] R. S. Smith and J. C. Doyle, “Model validation: A connection between robust
control and identification,” IEEE Trans. Auto. Control, vol. 37, pp. 942–952, July
1992.
[43] M. Newlin and R. S. Smith, “Model validation and generalized µ,” in Proc. IEEE
Control Decision Conf., pp. 1257–1258, 1991.
[44] K. Poolla, P. Khargonekar, A. Tikku, J. Krause, and K. Nagpal, “A time-domain
approach to model validation,” in Proc. Amer. Control Conf., pp. 313–317, 1992.
[45] R. S. Smith, “Model validation and parameter identification for systems in H and
∞
l1,” in Proc. Amer. Control Conf., pp. 2852–2856, 1992.
[46] T. Zhou and H. Kimura, “Input-output extrapolation-minimization theorem and its
application to model validation and robust identification,” in The Modeling of
Uncertainty in Control: Proceedings of the 1992 Santa Barbara Workshop (R. Smith
and M. Dahleh, eds.), pp. 127–137, Springer-Verlag, 1994.
[47] R. S. Smith, “Model validation for robust control: an experimental process control
application,” in Proc. of the 13th IFAC World Congress, vol. 9, pp. 61–64, July
1993.
[48] J. M. Krause and P. P. Khargonekar, “Parameter identification in the presence of
non-parametric dynamic uncertainty,” Automatica, vol. 26, pp. 113–124, 1990.
[49] R. Smith and J. Doyle, “Towards a methodology for robust parameter
identification,” in Proc. Amer. Control Conf., vol. 3, pp. 2394–2399, 1990.
Download from Www.Somanuals.com. All Manuals Search And Download.
[50] J. M. Krause, P. P. Khargonekar, and G. Stein, “Robust parameter adjustment
with nonparametric weighted-ball-in-H∞ uncertainty,” IEEE Trans. Auto. Control,
vol. AC-35, pp. 225–229, 1990.
[51] R. S. Smith and J. C. Doyle, “Closed loop relay estimation of uncertainty bounds
for robust control models,” in Proc. of the 13th IFAC World Congress, vol. 9,
pp. 57–60, July 1993.
[52] R. J. Schrama and P. M. V. den Hof, “An iterative scheme for identification and
control design based on coprime factorizations,” in Proc. Amer. Control Conf.,
pp. 2842–2846, 1992.
[53] R. J. P. Schrama, “Accurate identification for control: the necessity of an iterative
scheme,” IEEE Trans. Auto. Control, vol. 37, pp. 991–994, July 1992.
[54] Z. Zang, R. R. Bitmead, and M. Gevers, “H2 iterative model refinement and control
robustness enhancement,” in Proc. IEEE Control Decision Conf., pp. 279–284, 1991.
[55] D. Bayard, Y. Yam, and E. Mettler, “A criterion for joint optimization of
identification and robust control,” IEEE Trans. Auto. Control, vol. 37, pp. 986–991,
July 1992.
[56] Z. Zang, R. R. Bitmead, and M. Gevers, “Disturbance rejection: on-line refinement
of controllers by closed loop modelling,” in Proc. Amer. Control Conf.,
pp. 2929–2833, 1992.
[57] B. A. Francis, A Course in H Control Theory, vol. 88 of Lecture Notes in Control
∞
and Information Sciences. Berlin: Springer-Verlag, 1987.
[58] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State-space solutions to
standard H2 and H control problems,” IEEE Trans. Auto. Control, vol. AC-34,
∞
pp. 831–847, 1989.
[59] K. Glover and J. Doyle, “State-space formulae for all stabilizing controllers that
satisfy an H norm bound and relations to risk sensitivity,” Syst. and Control
∞
Letters, vol. 11, pp. 167–172, Oct 1988.
[60] B. D. O. Anderson, “An algebraic solution to the spectral factorization problem,”
IEEE Trans. Auto. Control, vol. AC-12, pp. 410–414, 1967.
[61] J. C. Willems, “Least-squares stationary optimal control and the algebraic Riccati
equation,” IEEE Trans. Auto. Control, vol. AC-16, pp. 621–634, 1971.
[62] S. Boyd, V. Balakrishnan, and P. Kabamba, “On computing the h norm of a
∞
transfer matrix,” Math Contr. Signals, Syst., 1988.
Download from Www.Somanuals.com. All Manuals Search And Download.
[63] A. J. Laub, “A Schur method for solving algebraic Riccati equations,” IEEE Trans.
Auto. Control, vol. AC-24, pp. 913–921, 1979.
[64] T. Pappas, A. J. Laub, and N. R. Sandell, “On the numerical solution of the
discrete-time algebraic Riccati equation,” IEEE Trans. Auto. Control, vol. AC-25,
pp. 631–641, 1980.
[65] W. F. Arnold and A. J. Laub, “Generalized eigenproblem algorithms and software
for algebraic Riccati equations,” Proc. IEEE, vol. 72, pp. 1746–1754, 1984.
[66] A. J. Laub, “Invariant subspace methods for the numerical solution of Riccati
Equations,” in The Riccati Equation (S. Bittanti, A. J. Laub, and J. C. Willems,
eds.), pp. 163–196, Springer-Verlag, Berlin, 1991.
[67] G. Zames, “On the input-output stability of nonlinear time-varying feedback
systems, parts I and II.,” IEEE Trans. Auto. Control, vol. AC-11, pp. 228–238 and
465–476, 1966.
[68] J. Doyle, “Analysis of feedback systems with structured uncertainties,” IEE
Proceedings, Part D, vol. 133, pp. 45–56, Mar. 1982.
[69] M. K. H. Fan and A. L. Tits, “Characterization and efficient computation of the
structured singular value,” IEEE Trans. Auto. Control, vol. AC-31, pp. 734–743,
1986.
[70] M. K. H. Fan and A. L. Tits, “m-form numerical range and the computation of the
structured singular value,” IEEE Trans. Auto. Control, vol. AC-33, pp. 284–289,
1988.
[71] J. C. Doyle, A. K. Packard, P. M. Young, R. S. Smith, and M. P. Newlin, “The
structured singular value,” Tech. Rep. NASA-CR-4524, NASA, March 1992.
[72] M. G. Safonov and J. Doyle, “Minimizing conservativeness of robust singular
values,” in Multivariable Control (S. Tzafestas, ed.), New York: Reidel, 1984.
[73] M. K. H. Fan, A. L. Tits, and J. C. Doyle, “Robustness in the presence of joint
parametric uncertainty and unmodeled dynamics,” in Proc. Amer. Control Conf.,
pp. 1195–1200, 1988.
[74] P. M. Young and J. C. Doyle, “Computation of the µ with real and complex
uncertainties,” in Proc. IEEE Control Decision Conf., pp. 1230–1235, 1990.
[75] P. M. Young, M. P. Newlin, and J. C. Doyle, “µ analysis with real parametric
uncertainty,” in Proc. IEEE Control Decision Conf., 1991.
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[76] M. Dahleh, A. Tesi, and A. Vicino, “Extremal properties for the parametric robust
performance problem,” Tech. Rep. UCSB-ME-91-4, Univ. California, Santa
Barbara, Mech. Eng., 1991. also submitted to 30th IEEE CDC.
[77] B. C. Moore, “Principal components analysis in linear systems: controllability,
observability and model reduction,” IEEE Trans. Auto. Control, vol. AC-26,
pp. 17–31, 1981.
[78] D. F. Enns, Model Reduction for Control System Design. PhD thesis, Stanford
University, 1984.
[79] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems
and their L∞-error bounds,” Int. J. of Control, vol. 39, no. 6, pp. 1115–1193, 1984.
[80] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. New Jersey:
Prentice-Hall, 1975.
[81] J. L. Adcock, “Curve fitter for pole-zero analysis,” Hewlett-Packard Journal, p. 33,
January 1987.
[82] M. Safonov, A. Laub, and G. Hartman, “Feedback properties of multivariable
systems: The role and use of the return difference matrix,” IEEE Trans. Auto.
Control, vol. 26, no. 1, 1981.
[83] G. Hartman, M. Barrett, and C. Greene, “Control designs for an unstable vehicle,”
Tech. Rep. NAS 4–2578, NASA Dryden Flight Research Center, 1979.
[84] P. Merkel and R. Whitmoyer, “Development and evaluation of precision control
modes for fighter aircraft,” in AIAA Guidance, Navigation and Cont. Conf., 1976.
Paper No. 76–1950.
[85] J. T. Spanos and M. C. O’Neal, “Nanometer level optical control on the JPL
Phase B testbed,” in ADPA/AIAA/ASME/SPIE Conf. Active Mat. & Adapt.
Struct., Nov 1991.
[86] M. C. O’Neal and J. T. Spanos, “Optical pathlength control in the nanometer
regime on the JPL Phase B interferometer testbed,” in SPIE Int. Symp. Optical
Appl. Sci. & Eng., July 1991.
[87] J. T. Spanos and A. Kissil, “Modeling and identification of the JPL Phase B
testbed,” in ADPA/AIAA/ASME/SPIE Conf. Active Mat. & Adapt. Struct., Nov
1991.
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Chapter 6
Function Reference
6.1 Xµ Functions
The following pages contain descriptions of the Xµ functions. These are also available
on-line via the help utility. Each description also gives an illustrative example of the
function’s use.
The functions are included in alphabetical order. For convenience they are
cross-referenced by typical use in the following list.
System building and interconnection
daug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
randsys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
starp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
sysic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Variable display and graphics
201
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rifd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
ctrlplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Time response calculations and pdm functions
gstep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
interp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
mergeseg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
randpdm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
sdtrsp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
trsp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
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Model reduction and state-space functions
balmoore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
modalstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
ophank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
orderstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
simtransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
sresidualize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
truncate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Controller synthesis
hinfsyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
h2syn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
hinfnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
h2norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
µ analysis and D-K iteration
blknorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
mkpert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
musynfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
spectrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
randpert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Transfer function fitting
fitsys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
mkphase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
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Miscellaneous functions
conpdm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
consys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
csum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
delsubstr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
substr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
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balmoore
Syntax
[SysR,HSV,T] = balmoore(Sys,{nsr,bound})
Parameter List
Inputs:
Sys
nsr
Linear, stable, minimal state-space system
(optional) If bound is used then a reduction will be per-
formed which meets an error bound specified by the value
in nsr, otherwise nsr is the order of the reduced system.
If nsr is not specified, the user will be prompted for its
value after the Hankel singular values are displayed.
Keywords: bound Boolean; meet an upper bound on the error (see nsr).
Outputs:
SysR
HSV
T
Internally balanced reduced order system; dynamic system
object.
A column vector containing the Hankel singular values of
the system, Sys.
Square matrix containing the balancing transformation,
i.e. xbal = Tx.
Description
Computes the balanced form of the system Sys which can be continuous or discrete,
then optionally truncates to the desired order, nsr, via B.C. Moore’s algorithm.
The user must ensure that the input system is minimal. Any initial state values or state
names associated with Sys are assigned to SysR. Input and Output names are also
maintained.
This function is cross-licensed from the Model Reduction Module.
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Reference
B.C. Moore, “Principal Component Analysis in Linear Systems: Controllability,
Observability and Model Reduction,” IEEE Trans. Auto. Ctrl., Vol. 26, No. 1, pp. 17–32,
Feb. 1981.
Example
# Create a five state system for reduction.
a = daug(-0.891334,[-1.20857,0.799042;-0.799042,-1.20857],...
-4.74685,-21.3013)
b = [0.0262569;-0.189601;-0.113729;0.211465;-0.538239]
c = [0.120725,-0.336942,0.397198,-0.700524,-1.02235]
d = 0
sys1 = system(a,b,c,d)
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
# Reduce to 3 states by balanced truncation.
[sysout1,hsv] = balmoore(sys1,{nsr=3})
sysout1g = freq(sysout1,fHz)
balerr = sys1g - sysout1g
# Reduce to a 3 state system by residualization
# for comparison purposes.
sysout2 = sresidualize(sys1,3)
sysout2g = freq(sysout2,fHz)
residerror = sys1g - sysout2g
gph1 = ctrlplot([sys1g,sysout1g,sysout2g,...
balerr,residerror],{logmagplot});
gph1 = plot(gph1,{!grid,legend=["original system";...
"reduced: balmoore";"reduced: sresidualize";...
"balmoore error";"sresidualize error"]})?
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See Also:
minimal, ophank.
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blknorm
Syntax
normM = blknorm(M,blk,p,Frobenius)
Parameter List
Inputs:
M
Matrix (or pdm).
blk
Block structure. See mu section of the manual for a de-
scription of the syntax.
p
Scalar valued. Specifies the Holder “p” norm to be used,
where 1 ≤ p ≤ inf. Optional. The default is p = 2.
Keywords: Frobenius The Frobenius norm is used.
Outputs: normM Matrix (or pdm) norms of each block of M.
Description
M is partitioned according the input/output partitions determined by blk. The
maximum singular value of each partition is calculated and normA is the matrix (pdm)
of norms.
For example, if p = 2 (the default case) and blk represents a single perturbation (with
compatible input/output dimensions) then normA is simply the maximum singular
value. If blk consists of entirely 1×1 perturbations, then normA is equal to abs(A).
Repeated scalar blocks are taken as full blocks. This neglects the assumed equivalences
between parts of Delta.
The Xmath function norm is used for the calculations. For further detail, refer to norm
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Examples
A = random(3,3)-0.5*ones(3,3)?
A (a square matrix) =
0.0618661
0.390622
-0.112622
0.0896177
0.00422128
0.42229
0.185398
-0.150638
0.448818
blknorm(A,[1,1; 1,1; 1,1])
ans (a square matrix) =
0.0618661
0.390622
0.112622
0.0896177
0.00422128
0.42229
0.185398
0.150638
0.448818
blknorm(A,[3,3])
ans (a scalar) = 0.682429
# compare to the following:
max(svd(A))
ans (a scalar) = 0.682429
B = [1,2,3,4; 5,6,7,-8; 9,10,-11,12]?
B (a rectangular matrix) =
1
5
9
2
6
10
3
7
-11
4
-8
12
blk = [2,1;1,1;1,1]
# examine 1,1 entry of blknorm result
blknorm(B,blk)
ans (a square matrix) =
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2.23607
7.81025
13.4536
3
7
11
4
8
12
# and compare to
norm(B(1,1:2))
ans (a scalar) = 2.23607
See Also
norm
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conpdm
Syntax
outpdm = conpdm(mat,domain,{skipChks})
Parameter List
Inputs:
mat
constant matrix
domain
domain over which outsys will be defined.
Keywords: skipChks Boolean specifying that syntax checking is to be skipped.
Outputs: outpdm pdm
Description
Creates a pdm data object from a constant matrix. Outpdm represents a constant gain;
its value is repeated at every instance of the domain. It is equivalent to,
outpdm = pdm(kronecker(ones(length(dom),1),mat),dom)
and is a useful shorthand for including constant values in a pdm plot.
This function may be superceded by redefining the augmentation operators
and other functions in a later release of Xmath. To maintain future upwards
compatibility avoid using this function when developing derivative software.
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consys
Syntax
outsys = consys(mat,{skipChks})
Parameter List
Inputs:
Keywords: skipChks Boolean specifying that syntax checking is to be skipped.
Outputs: outsys Dynamic System
mat
constant matrix
Description
Creates a Dynamic System object from a constant matrix. Outsys represents a
constant gain; the A, B and C matrices are empty. It is equivalent to,
outsys = system([],[],[],mat)
and can be useful in specifying constant inputs to freq.
This function may be superceded by redefining the augmentation operators
and other functions in a later release of Xmath. To maintain future upwards
compatibility avoid using this function when developing derivative software.
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csum
Syntax
[outpdm] = csum(inpdm, {channels})
Parameter List
Inputs:
inpdm
real or complex valued pdm or constant matrix
Keywords: channels Sum over channels. outpdm has the same dimensions as
inpdm.
Outputs:
outpdm output pdm
Description
Perform a cumulative sum over the rows of a matrix or a pdm. If channels is specified
then the sum is performed over the domain of the pdm.
Examples
A = [ones(6,1),random(6,1)]?
A (a rectangular matrix) =
1
1
1
1
1
1
0.608453
0.854421
0.0642647
0.827908
0.926234
0.566721
csum(A)
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ans (a rectangular matrix) =
1
2
3
4
5
6
0.608453
1.46287
1.52714
2.35505
3.28128
3.848
pdmA = pdm(A,[1,2,3])?
pdmA (a pdm) =
domain |
Col 1 Col 2
-------+------------------------
1 | Row 1
| Row 2
1
1
0.608453
0.854421
-------+------------------------
2 | Row 1
| Row 2
1
1
0.0642647
0.827908
-------+------------------------
3 | Row 1
| Row 2
1
1
0.926234
0.566721
-------+------------------------
csum(pdmA)
ans (a pdm) =
domain |
Col 1 Col 2
-------+------------------------
1 | Row 1
| Row 2
1
2
0.608453
1.46287
-------+------------------------
2 | Row 1
| Row 2
1
2
0.0642647
0.892173
-------+------------------------
3 | Row 1
| Row 2
1
2
0.926234
1.49296
-------+------------------------
csum(pdmA,channels)
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ans (a pdm) =
domain |
Col 1 Col 2
-------+-----------------------
1 | Row 1
| Row 2
1
1
0.608453
0.854421
-------+-----------------------
2 | Row 1
| Row 2
2
2
0.672717
1.68233
-------+-----------------------
3 | Row 1
| Row 2
3
3
1.59895
2.24905
-------+-----------------------
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ctrlplot
Syntax
graph = ctrlplot(pdm,old graph,{keywords})
Parameter List
Inputs:
pdm
Pdm (or matrix) containing the data to be plotted.
old graph
(optional) Graphical object to which data is added. Con-
ceptually the same as plot(pdm,{keep=old graph}).
Keywords:
The following keywords specify the basic plot format. Only one can be selected.
timeresp
bode
(default) real(pdm) vs. domain. This is the same as the
default plot function and is suitable for time domain re-
sponses
Two subplots are generated: log magnitude vs. domain
and angle vs. domain. They are positioned one above the
other. If present, old graph must also be in this format.
nyquist
imag(pdm) vs. real(pdm). Standard Nyquist plot.
log magnitude vs. angle. Standard Nichols chart.
log magnitude vs. domain.
nichols
logmagplot
phaseplot
angle vs. domain.
The following keywords specify whether the domain is log or linear scale. This is not
applicable to the Nyquist or Nichols plots. The defaults depend upon which of the above
control plot types has been selected.
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linear
log
linear domain. Default = 1 for timeresp keyword. Default
= 0 for bode keyword.
logarithmic domain. Default = 1 for bode keyword. De-
fault = 0 for timeresp keyword.
Default units can be supplied for the magnitude and phase plots (bode, nichols,
logmagplot and phaseplot keywords) with the following keywords.
degrees
db
Angles are specified in degrees (default = 1)
log magnitudes are specified in decibels (default = 0).
The following keywords behave identically to those in plot. They relate directly to the
line/marker specifications and must be associated with a particular pdm.
line
(default = 1). Plot as a line type.
(default = 0). Plot markers.
refer to plot
marker
line style
line width
refer to plot
marker style refer to plot
marker size refer to plot
Some preprocessing of the data is performed by the following keyword.
unwrap
Applicable to bode, nichols and phaseplot keywords.
Phase changes of 2π are unwrapped, rather than being
graphed between −π and π. Default = 1.
Outputs:
graph
Resulting graphical object.
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Description
This function performs some common control system related plotting. The user can use
ctrlplot to set up a basic plot and perform some preprocessing of the data. This
generates a graphical object containing the data and the user can perform subsequent
calls to plot to add things like text, labels, gridding etc. The second argument
(optional) is a graphical object to which the pdm will be added.
Plots over a domain (bode, timeresp, logmagplot, phaseplot) must be called with a pdm.
A matrix or scalar can be turned into a suitable pdm with the conpdm function. Nyquist
and Nichols plots can also plot scalars. Points such as -1 are often useful. The user
should specify a marker so that such points show up. When mixing scalar and pdm data
on a Nyquist or Nichols plot, plot the pdm data first to get a reasonable choice of axes.
For Bode plots, two graphs are created — the magnitude and the phase plots — and
positioned one above the other. The result is returned as a single graphical object.
Similarly, when adding data to Bode plots, the existing graphical object must also
contain two subplots.
Default labels, corresponding to the most common use of the particular invocation, are
put on the axes. These can be overwritten with subsequent calls to plot.
Examples:
# Create 2 systems.
sys1 = 1/makepoly([1,1],"s")
sys2 = 2*sys1*10/makepoly([1,1,10],"s")
w1 = logspace(0.01,10,50)’
w2 = sort([w1;[0.35:0.01:0.65]’])
sys1g = freq(sys1,w1)
sys2g = freq(sys2,w2)
# Bode plots
g1 = ctrlplot(sys1g,{bode});
g1 = ctrlplot(sys2g,g1,{bode});
g1 = plot({keep=g1,title = "Bode plots",...
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legend = ["sys1","sys2"]})?
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Bode plots
10
1
0.1
sys1
sys2
0.01
0.001
0.0001
1e-05
0.01
0.1
1
10
Frequency
0
-50
-100
-150
-200
-250
-300
0.01
0.1
1
10
Frequency
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# Nyquist plots
g2 = ctrlplot(sys1g,{nyquist});
g2 = ctrlplot(sys2g,g2,{nyquist});
g2 = ctrlplot(-1,g2,{nyquist,marker=1,line=0});
g2 = plot(g2,{projection="orthographic",...
legend=["sys1","sys3","critical point"],title="Nyquist plots"})?
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Nyquist plots
0.5
0
-0.5
sys1
sys3
critical point
-1
-1.5
-2
-2
-1
0
1
2
Real
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# Create a second order lightly damped system to illustrate
# time response plotting. The calculation is repeated with
# a non-zero initial condition.
sys = 5/makepoly([1,1,5],"s")
u = gstep([0:0.05:10],[0;1;5],[0;1;-1])
y0 = sys*u
[a,b,c,d] = abcd(sys)
sys = system(a,b,c,d)
y1 = system(sys,{X0=[-1;0]})*u
# Now plot the result
g1 = ctrlplot(u,{line style=2});
g1 = ctrlplot(y0,g1,{line style=1});
g1 = ctrlplot(y1,g1,{line style=4});
g1 = plot(g1,{!grid,legend=["input";"x0 = zero";"non-zero x0"]})?
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2
1
0
input
-1
-2
-3
x0 = zero
non-zero x0
0
2
4
6
8
10
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See Also:
plot.
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daug
Syntax
out = daug (sys1,sys2,...)
Parameter List
Inputs:
sys1 Input systems. These can be dynamical systems and con-
stants, or pdms and constants.
.
.
.
“
Outputs: out
output system.
Description
Diagonal augmentation of dynamical system/pdm/constant, matrices.
sys1
0
. . .
0
0
0
.
.
.
sys2 . . .
.
.
.
out =
.
.
.
.
.
.
0
0
. . . sysN
Limitations
Only 21 systems can be augmented with a single function invocation.
Examples
daug([1,1],[2;2],inf)
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ans (a square matrix) =
1
0
0
0
1
0
0
0
0
2
2
0
0
0
0
Inf
sys1 = randsys(1,1,2,{stable})
sys1 = system(sys1,{statenames="sys1state"})?
sys1 (a state space system) =
A
-0.886949
B
0.853282
0.012459
0.748961
C
0.186754
D
0.492058
X0
0
State Names
-----------
sys1state
System is continuous
sys2 = randsys(1,2,1,{stable})
sys2 = system(sys2,{statenames="sys2state",x0=1})?
sys2 (a state space system) =
A
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-1.67106
B
0.579502
C
0.262815
0.436099
D
0.911055
0.808267
X0
1
State Names
-----------
sys2state
System is continuous
# Note the effect of the constant in the following
daug(sys1,10,sys2)
ans (a state space system) =
A
-0.886949
0
0
-1.67106
B
0.853282
0
0.012459
0
0
0
0
0.579502
C
0.186754
0
0
0
0
0
0.262815
0.436099
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D
0.492058
0.748961
0
10
0
0
0
0
0
0
0
0
0
0.911055
0.808267
0
X0
0
1
State Names
-----------
sys1state
sys2state
System is continuous
pdm1 = randpdm(3,2,2)?
pdm1 (a pdm) =
domain |
Col 1
Col 2
-------+---------------------------
0 | Row 1 0.810265 0.259043
| Row 2 0.413909 0.359993
-------+---------------------------
1 | Row 1 0.691279 0.765686
| Row 2 0.357265 0.76934
-------+---------------------------
2 | Row 1 0.547763 0.0962289
| Row 2 0.956117 0.220741
-------+---------------------------
pdm2 = pdm(10*ones(3,1),domain(pdm1))?
pdm2 (a pdm) =
domain |
-------+-----
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0 |
-------+-----
1 | 10
-------+-----
2 | 10
10
-------+-----
daug(pdm1,pdm2)
ans (a pdm) =
domain |
Col 1
Col 2
Col 3
-------+----------------------------------
0 | Row 1 0.810265 0.259043
| Row 2 0.413909 0.359993
0
0
| Row 3
0
0
10
-------+----------------------------------
1 | Row 1 0.691279 0.765686
| Row 2 0.357265 0.76934
0
0
| Row 3
0
0
10
-------+----------------------------------
2 | Row 1 0.547763 0.0962289
| Row 2 0.956117 0.220741
0
0
| Row 3
0
0
10
-------+----------------------------------
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delsubstr
Syntax
[outstr] = delsubstr(str,charstr)
Parameter List
Inputs:
str
String or vector of strings.
charstr String
Outputs: outstr
String or vector of strings.
Description
All occurences of the substring, charstr, within str are deleted.
If, by deleting charstr, another occurence of charstr in created, it will not be deleted.
Examine the second example closely to see the effect of this.
Unless str is a scalar string, deleting a whole string element will cause an error.
Example
strvec = ["string one";"aaa";"xxyy"]
out1 = delsubstr(strvec,"g o")?
out1 (a column vector of strings) =
strinne
aaa
xxyy
out2 = delsubstr(strvec,"xy")?
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out2 (a column vector of strings) =
string one
aaa
xy
# If executed, the following would give an error
# out3 = delsubstr(strvec,"a")?
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fitsys
Syntax
[sys] = fitsys(data,npoles,nzeros,weight, {skipchks,Hertz})
Parameter List
Inputs:
data
Complex valued data (pdm).
npoles
nzeros
Order of requested fit. (optional, default = 0).
Number of zeros in transfer function. (optional, default =
npoles)
weight
Weighting function. (scalar, pdm, or Dynamic System)
(optional, default = 1).
Keywords: Hertz
Boolean. This keyword is mandatory as the function must
know whether the domain is in Hertz or radians/second
(specified by !Hertz) to fit correctly. Note that the Xmath
function freq assumes that the frequency range is specified
in Hertz.
skipchks Boolean. Skip the error checking. (Default = 0)
Outputs:
sys Dynamic system, order = npoles.
Description
Fits a transfer function to complex valued data. npoles and nzeros specifies the number
of poles and zeros.
The optional argument weight, specifies a weighting for the fit. If weight is a pdm it
must be over the same domain as data. It may also be a Dynamic System, in which
case the magnitude of its frequency response is the weight. A scalar weight may also be
specified although this will have no effect. For logscale frequency data a weight of close
to 1/s is strongly recommended.
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The primary use of this routine is the fitting of D scale weights for mu synthesis
iterations.
Chebyshev polynomials are used as basis functions for both the numerator and
denominator polynomials.
WARNING: This routine uses iterative polynomial calculations which are not well
conditioned for high order (> 6) fits.
Reference
For further information see: “Curve Fitter for Pole-Zero Analysis,” J.L. Adcock,
Hewlett-Packard Journal, p. 33, January 1987.
Example
# Set up a plant to generated data for the fitting
# problem.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Note that plant has right half plane zeros
rifd(plant)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-5.0000e+00
3.1225e-01
-3.1225e-01
0.0000e+00
3.1623e-01
3.1623e-01
5.0000e+00
0.1581
0.1581
1.0000
Zeros:
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real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0000e+00
5.0000e-01
5.0000e-01
0.0000e+00
3.1225e+00
-3.1225e+00
1.0000e+00
3.1623e+00
3.1623e+00
1.0000
-0.1581
-0.1581
omega = logspace(0.001,100,200)
plantg = freq(plant,omega)
# Use complex cepstrum to fit minimum phase equivalent
# to the magnitude of the data. One of the principle
# uses of the fitsys function is fitting approximations
# to noisy data. To illustrate the concepts, no noise
# is added here.
cdata = mkphase(abs(plantg))
gph1 = ctrlplot([plantg,cdata],{phaseplot});
gph1 = plot(gph1,{title="Data and minimum phase fit",...
legend=["original data";"minimum phase fit"]})?
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Data and minimum phase fit
100
0
-100
-200
-300
-400
original data
minimum phase fit
0.001
0.01
0.1
1
10
100
Frequency
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# Create fitting weight. 1/s works well for logspaced
# data.
wght = 1/makepoly([1,0],"s")
# Fit new system and compare pole location with
# the original. Note that it is minimum phase.
nsys = fitsys(cdata,3,3,wght)
rifd(nsys)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.1043e-02
-5.1043e-02
-5.0107e+00
-3.1249e-01
3.1249e-01
0.0000e+00
3.1663e-01
3.1663e-01
5.0107e+00
0.1612
0.1612
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0189e+00
-5.0906e-01
-5.0906e-01
0.0000e+00
-3.0989e+00
3.0989e+00
1.0189e+00
3.1405e+00
3.1405e+00
1.0000
0.1621
0.1621
nsysg = freq(nsys,omega)
gph2 = ctrlplot([plantg,nsysg],{bode});
gph2 = plot(gph2,{title="Data and minimum phase fit",...
legend=["original system";"minimum phase system"]})?
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Data and minimum phase fit
100
10
original system
minimum phase system
1
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
100
0
-100
-200
-300
-400
0.001
0.01
0.1
1
10
100
Frequency
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Limitations
Limited to SISO systems.
See Also
tfid
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gstep
Syntax
gPdm = gstep (ytime,timespec,valspec, {skipChks})
Parameter List
Inputs:
ytime
timespec times for specified step data (optional)
valspec value for specified step data (optional)
output time vector (seconds).
Keywords: skipChks Boolean specifying that syntax checking is to be skipped.
Outputs: gPdm pdm containing the step values as a function of time.
Description
This function creates a PDM over the domain: ytime. At ytime = timespec(i) the
output steps to value: valspec(i) and maintains that value until the next specified step
or the end of the domain.
Example
time = [0:100]
steptimes = [5,25,30,65,90]
stepvalues = [-1,2,1,-1.5,1.5]
out = gstep(time,steptimes,stepvalues)
gph1 = ctrlplot(out);
gph1 = plot(gph1,{title="gstep example"})?
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Data and minimum phase fit
100
10
original system
minimum phase system
1
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
100
0
-100
-200
-300
-400
0.001
0.01
0.1
1
10
100
Frequency
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See Also
randpdm, gcos, gsin, gpulse, gsawtooth, gsquarewave
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hinfnorm
Syntax
[out,omega] = hinfnorm(sys,tol,{imag eps,max it})
Parameter List
Inputs:
sys
sys
Dynamic System, frequency response (pdm), or constant
gain (matrix).
Specifies the relative tolerance of the answer when the in-
put is a Dynamic SystemDefault = 0.001.
Keywords: imag eps Epsilon value for determining imaginary eigenvalues of the
Hamiltonian. Default = sqrt(eps).
max it
Maximum number of iterations. Default = 100.
Outputs:
out
H norm of the input system. Scalar or pdm or matrix
∞
input systems. 2×1 vector for Dynamic System inputs.
An upper bound of inf indicates that the maximum number
of iterations was exceeded.
omega
Frequency (Hz) where the norm is achieved.
Description
The H norm is defined to be the supremum, over frequency, of the maximum singular
∞
value of the system’s frequency response.
If the input system is a pdm, it is assumed to be a frequency response and the norm is
calculated only from the frequencies provided.
If the inputs system is a matrix, it represents a constant gain and the maximum singular
value of the matrix is returned. Note that this is NOT the same as the result of
norm(sys,inf).
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Stable Dynamic System norms are calculated by an iterative Hamiltonian method. In
this case out is a 2×1 vector with upper and lower bounds for the norm.
Example
# Set up a simple closed loop problem.
# This example is given in more detail in the
# hinfsyn online help.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Create weights
Wperf = 100/makepoly([100,1],"s")
Wact = makepoly([0.5,0.05],"s")/makepoly([0.05,1],"s")
# Form the weighted interconnection structure
sysnames = ["plant";"Wperf";"Wact"]
sysinp = ["ref";"control"]
sysout = ["Wperf"; "Wact"; "ref-plant"]
syscnx = ["control"; ...
"ref-plant"; ...
"control"]
# input to plant
# input to Wperf
# input to Wact
wghtic = sysic(sysnames,sysinp,sysout,syscnx,plant,...
Wperf,Wact)
# Design Hinf controller
nctrls = 1
nmeas = 1
gmax = 25
gmin = 0
Kinf = hinfsyn(wghtic,nmeas,nctrls,[gmax;gmin])
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
25.0000
Y eig
Hx eig
X eig
nrho xy
p/f
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25.000
12.500
6.250
3.125
1.562
0.781
1.172
5.2e-01
5.2e-01
5.2e-01
5.1e-01
5.0e-01
1.7e-03
1.7e-03
1.7e-03
1.7e-03
1.7e-03
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
p
p
p
p
p
f
p
3.9e-01 -9.8e+01
4.8e-01 1.8e-03
Gamma value achieved:
1.1719
# form weighted closed loop system
clpinf = starp(wghtic,Kinf)
# Compare hinfnorm to gamma value. The H infinity
# norm should be less than or equal to gamma.
hinfnorm(clpinf)?
ans (a column vector) =
1.1701
1.16893
See Also
h2norm, hinfsyn, h2syn.
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h2norm
Syntax
out = h2norm(sys)
Parameter List
Inputs:
sys Continuous time Dynamic System
Outputs: out H2 norm of the input system
Description
The H2 norm of a stable, strictly proper system is calculated. This is given by
out = trace(CXC0),
where X is the controllability grammian, solving the Lyapunov equation,
AX + XA0 + BB0 = 0.
Example
# Set up a simple closed loop problem.
# This example is also studied in the
# hinfsyn on-line help.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Create weights
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Wperf = 100/makepoly([100,1],"s")
Wact = makepoly([0.5,0.05],"s")/makepoly([0.05,1],"s")
# Form the weighted interconnection structure
sysnames = ["plant";"Wperf";"Wact"]
sysinp = ["ref";"control"]
sysout = ["Wperf"; "Wact"; "ref-plant"]
syscnx = ["control"; ...
"ref-plant"; ...
"control"]
# input to plant
# input to Wperf
# input to Wact
wghtic = sysic(sysnames,sysinp,sysout,syscnx,plant,...
Wperf,Wact)
# Design H2 controller
nctrls = 1
nmeas = 1
K2 = h2syn(wghtic,nmeas,nctrls)
# Form the weighted closed loop system and calculate
# its H2 norm.
wghtclp2 = starp(wghtic,K2)
h2norm(wghtclp2)?
ans (a scalar) =
2.78655
See Also
h2syn, hinfsyn, hinfnorm
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hinfsyn
Syntax
[k,gfin,stat] = hinfsyn(p,nmeas,ncon,gamma,{keywords})
Parameter List
Inputs:
p
Generalized interconnection structure (Dynamic Sys-
tem)
nmeas
measurement vector dimension.
ncon
gamma
control vector dimension.
H
∞
norm bound of controller. For a bisection search spec-
ify gamma = [gamma min;gamma max].
Keywords: schur solution real Schur decomposition for Riccati solution (default)
eig solution
tol
eigendecomposition for Riccati solution.
tol=value. Specifies the relative tolerance for stopping a
bisection fit. Default = (gamma max - gamma min)/50
epr
epr = value. Tolerance for determining when the Hamilto-
nian eigenvalues lie on the jω axis. Default = 0.5*sqrt(eps)
epp
epp = value. Tolerance for determining that a Riccati
equation solution is positive definite. Default = 1e-6
maxit = value. Maximum number of bisection iterations.
Default = inf.
maxit
Outputs:
k
Central H optimal controller.
∞
gfin
stat
H
norm achieved for the returned controller.
∞
return status
0
Controller calculated
-1 no controller exists for specified gamma value.
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Description
The H (sub)optimal controller for the interconnection, p, is calculated. The resulting
∞
closed loop system is illustrated below.
z
w
p
y
u
-
k
The variables ncon and nmeas are used to specify the dimensions of u and y in the above
diagram (ncon = dim(u) and nmeas = dim(y)). The objective is to design a stabilizing
controller, k, which minimizes the H2 norm of the closed loop system between w and z.
The closed loop system can be formed with the command,
clpsys = starp(p, k).
p is a state-space system, which can be partitioned with respect to [z; y] and [w; u] in
the following way.
a
b1
b2
p = c1 d11 d12
c2 d21 d22
.
The following assumptions must hold:
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1. (a, b2, c2) is stabilizable and detectable
2. d12 and d21 have full rank
3. The matrix [a − jωI, b2; c1, d12] has full column rank for all ω
4. The matrix [a − jωI, b1; c2, d21] has full row rank for all ω
Reference
This function uses the state-space formulae given in:
“State-space formulae for all stabilizing controllers that satisfy an H norm bound and
∞
relations to risk sensitivity,” Keith Glover and John Doyle, Systems & Control Letters
11, pp. 167–172., Oct, 1988.
Example
# Set up a simple closed loop problem.
# A tracking problem is chosen. Weights are used to
# trade off between tracking performance and actuator
# effort.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Create weights (performance & actuator)
Wperf = 100/makepoly([100,1],"s")
Wact = makepoly([0.5,0.05],"s")/makepoly([0.05,1],"s")
# Form the weighted interconnection structure
sysnames = ["plant";"Wperf";"Wact"]
sysinp = ["ref";"control"]
sysout = ["Wperf"; "Wact"; "ref-plant"]
syscnx = ["control"; ...
"ref-plant"; ...
"control"]
# input to plant
# input to Wperf
# input to Wact
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wghtic = sysic(sysnames,sysinp,sysout,syscnx,plant,...
Wperf,Wact)
# Design Hinf controller
nctrls = 1
nmeas = 1
gmax = 25
gmin = 0
Kinf = hinfsyn(wghtic,nmeas,nctrls,[gmax;gmin])
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
25.0000
Y eig
Hx eig
X eig
nrho xy
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
p/f
25.000
12.500
6.250
3.125
1.562
5.2e-01
5.2e-01
5.2e-01
5.1e-01
5.0e-01
1.7e-03
1.7e-03
1.7e-03
1.7e-03
1.7e-03
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
1.0e-02
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
0.0e+00
p
p
p
p
p
f
p
0.781
1.172
3.9e-01 -9.8e+01
4.8e-01 1.8e-03
Gamma value achieved:
rifd(Kinf)
1.1719
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0000e-02
-1.0347e+00
-2.6846e+00
-2.6846e+00
-1.2692e+01
0.0000e+00
1.0000e-02
1.0347e+00
3.5362e+00
3.5362e+00
1.2692e+01
1.0000
1.0000
0.7592
0.7592
1.0000
0.0000e+00
2.3017e+00
-2.3017e+00
0.0000e+00
Zeros:
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real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-5.0000e+00
-2.0000e+01
-3.1225e-01
3.1225e-01
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
1.0000
omega = logspace(0.001,100,200)
Kinfg = freq(Kinf,omega)
gph1 = ctrlplot(Kinfg,{bode});
gph1 = plot(gph1,{title="Kinf"})?
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Kinf
10
1
0.1
0.01
0.001
0.001
0.01
0.1
1
10
100
Frequency
40
20
0
-20
-40
-60
-80
-100
0.001
0.01
0.1
1
10
100
Frequency
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# Use sysic to create unweighted interconnection
ic = sysic("plant",["ref";"ctrl"],["plant";"ref-plant"],...
"ctrl",plant)
clpinf = starp(ic,Kinf)
rifd(clpinf)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-9.5105e-01
-9.7350e-01
-9.7350e-01
-5.0000e+00
-5.0901e+00
-1.1812e+01
3.1225e-01
-3.1225e-01
0.0000e+00
-1.2285e+00
1.2285e+00
0.0000e+00
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
9.5105e-01
1.5675e+00
1.5675e+00
5.0000e+00
5.0901e+00
1.1812e+01
0.1581
0.1581
1.0000
0.6211
0.6211
1.0000
1.0000
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-1.0000e+00
5.0000e-01
5.0000e-01
-5.0000e+00
-2.0000e+01
3.1225e-01
-3.1225e-01
0.0000e+00
3.1225e+00
-3.1225e+00
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
1.0000e+00
3.1623e+00
3.1623e+00
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
-0.1581
-0.1581
1.0000
1.0000
# Examine sensitivity function
sens = inv(1 + plant*Kinf)
sensg = freq(sens,omega)
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gph2 = ctrlplot(sensg,{logmagplot});
gph2 = plot(gph2,{title="Kinf controller: sensitivity function"})?
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Kinf controller: sensitivity function
10
1
0.1
0.01
0.001
0.01
0.1
1
10
100
Frequency
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# Examine step response
step = gstep([0:0.1:10],0,1)
y = clpinf*step
gph3 = ctrlplot([y,step]);
gph3 = plot(gph3,{title="Kinf controller: step response"})?
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Kinf controller: step response
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
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See also
hinfsyn, hinfnorm, h2norm
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h2syn
Syntax
k = h2syn(p,nmeas,ncon,{keywords})
Parameter List
Inputs:
p
Generalized interconnection structure (Dynamic Sys-
tem)
nmeas
ncon
measurement vector dimension.
control vector dimension.
Keywords: schur solution real Schur decomposition for Riccati solution (default)
eig solution
epr
eigendecomposition for Riccati solution.
epr = value. Tolerance for determining when the Hamilto-
nian eigenvalues lie on the jω axis. Default = 0.5*sqrt(eps)
Outputs:
k
H2 optimal controller.
Description
The H2 optimal controller for the interconnection, p, is calculated. The resulting closed
loop system is illustrated below.
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z
w
p
y
u
-
k
The variables ncon and nmeas are used to specify the dimensions of u and y in the above
diagram (ncon = dim(u) and nmeas = dim(y)). The objective is to design a stabilizing
controller, k, which minimizes the H2 norm of the closed loop system between w and z.
p is a state-space system, which can be partitioned with respect to [z; y] and [w; u] in
the following way.
a
b1
b2
p = c1 d11 d12
c2 d21 d22
.
The following assumptions must hold:
1. (a, b2, c2) is stabilizable and detectable
2. d12 and d21 have full rank
3. d11 = 0
4. The matrix [a − jωI, b2; c1, d12] has full column rank for all ω
5. The matrix [a − jωI, b1; c2, d21] has full row rank for all ω
In theory the Riccati equations should always have a solution. However, Hamiltonian
eigenvalues which are close to the imaginary axis will give problems. epr specifies a
tolerance for how close the calculated eigenvalues can be to the jω axis.
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Reference
This function uses the state-space formulae given in:
“State-space formulae for all stabilizing controllers that satisfy an H norm bound and
∞
relations to risk sensitivity,” Keith Glover and John Doyle, Systems & Control Letters
11, pp. 167–172., Oct, 1988.
Example
# Set up a simple closed loop problem.
# This example is also studied in the
# hinfsyn on-line help.
plant = makepoly([0.1,-0.1,1],"s")*makepoly([1,1],"s")...
/(makepoly([1,0.1,.1],"s")*(makepoly([0.2,1],"s")))
# Create weights. These are definitely not the best
# for an H2 design - they are chosen to match with
# the Hinf design example for comparison purposes.
Wperf = 100/makepoly([100,1],"s")
Wact = makepoly([0.5,0.05],"s")/makepoly([0.05,1],"s")
# Form the weighted interconnection structure
sysnames = ["plant";"Wperf";"Wact"]
sysinp = ["ref";"control"]
sysout = ["Wperf"; "Wact"; "ref-plant"]
syscnx = ["control"; ...
"ref-plant"; ...
"control"]
# input to plant
# input to Wperf
# input to Wact
wghtic = sysic(sysnames,sysinp,sysout,syscnx,plant,...
Wperf,Wact)
# Design H2 controller
nctrls = 1
nmeas = 1
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K2 = h2syn(wghtic,nmeas,nctrls)
rifd(K2)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.4046e-01
-1.4046e-01
-1.5863e+00
-1.5863e+00
-5.2060e+00
-2.3161e-01
2.3161e-01
3.4754e+00
-3.4754e+00
0.0000e+00
2.7087e-01
2.7087e-01
3.8203e+00
3.8203e+00
5.2060e+00
0.5186
0.5186
0.4152
0.4152
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-5.0000e+00
-2.0000e+01
-3.1225e-01
3.1225e-01
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
1.0000
omega = logspace(0.001,100,200)
K2g = freq(K2,omega)
gph1 = ctrlplot(K2g,{bode});
gph1 = plot(gph1,{title="K2"})?
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K2
1
0.1
0.01
0.001
0.0001
0.001
0.01
0.1
1
10
100
Frequency
50
0
-50
-100
-150
0.001
0.01
0.1
1
10
100
Frequency
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# Use sysic to create unweighted interconnection
ic = sysic("plant",["ref";"ctrl"],["plant";"ref-plant"],...
"ctrl",plant)
clp2 = starp(ic,K2)
rifd(clp2)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-1.8120e-01
-1.8120e-01
-1.6753e+00
-1.6753e+00
-4.9957e+00
-5.0000e+00
3.1225e-01
-3.1225e-01
-4.2333e-01
4.2333e-01
3.4263e+00
-3.4263e+00
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
4.6048e-01
4.6048e-01
3.8139e+00
3.8139e+00
4.9957e+00
5.0000e+00
0.1581
0.1581
0.3935
0.3935
0.4393
0.4393
1.0000
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e-02
-5.0000e-02
-1.0000e+00
5.0000e-01
5.0000e-01
-5.0000e+00
-2.0000e+01
3.1225e-01
-3.1225e-01
0.0000e+00
3.1225e+00
-3.1225e+00
0.0000e+00
0.0000e+00
3.1623e-01
3.1623e-01
1.0000e+00
3.1623e+00
3.1623e+00
5.0000e+00
2.0000e+01
0.1581
0.1581
1.0000
-0.1581
-0.1581
1.0000
1.0000
# Examine sensitivity function
sens = inv(1 + plant*K2)
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sensg = freq(sens,omega)
gph2 = ctrlplot(sensg,{logmagplot});
gph2 = plot(gph2,{title="K2 controller: sensitivity function"})?
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K2 controller: sensitivity function
10
1
0.1
0.001
0.01
0.1
1
10
100
Frequency
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# Examine step response
step = gstep([0:0.1:10],0,1)
y = clp2*step
gph3 = ctrlplot([y,step]);
gph3 = plot(gph3,{title="K2 controller: step response"})?
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K2 controller: step response
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
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See also
hinfsyn, h2norm, hinfnorm
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interp
Syntax
outpdm = interp(inpdm,stepsize,final {keywords})
outpdm = interp(inpdm,domspec, {keywords})
Parameter List
Inputs:
inpdm
stepsize
final
Input pdm.
Increment in outpdm domain.
Last value in outpdm domain. Optional: default =
max(domain(inpdm)).
domspec Regular vector or pdm used to specify the domain of out-
pdm.
Keywords: order
Interpolation order. Values are:
0
1
zero order hold (default)
linear interpolation
Outputs:
outpdm
interpolated pdm
Description
m is interpolated to give outpdm. Two syntaxes are available for specifying the domain
of outpdm.
In the first the domain is,
[min(domain(inpdm)):final:stepsize].
If final is not specified max(domain(inpdm)) is used.
The second syntax is, domain(domspec) if domspec is a pdm or domspec if domspec is a
vector.
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This function differs from the interpolate function in that it can handle zero order
hold type interpolation and deal with irregularly spaced input pdms. Irregularly spaced
output pdms can be generated with the domspec syntax. These features are often useful
when dealing with data generated from experiments.
Example
time = [0:1:5]
u = gstep(time,time,time)
u1 = interp(u,0.25,{order=0})
time2 = sort(5*random(20,1))
u2 = interp(u,time2,{order=1})
gph1 = ctrlplot(u2,{marker=1,marker style=1,...
marker size=1,line=0}); gph1 =
plot(u1,gph1,{marker=1,marker style=6,...
marker size=1,line=0}); gph1 =
plot(u,gph1,{marker=1,marker style=9,...
marker size=1,line=0}); gph1 = plot(gph1,{legend=["1st
order interp.";...
"0 order interp.";"original pdm"]})?
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5
4
3
2
1
0
1st order interp.
0 order interp.
original pdm
0
1
2
3
4
5
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See Also
interpolate
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mergeseg
Syntax
outpdm = mergeseg(pdm1,pdm2, {keywords})
Parameter List
Inputs:
pdm1
pdm2
input pdm
input pdm
Keywords: domsort
Sort the result of merging the pdms. If !domsort then
pdm2 is simply concatenated onto pdm1. Boolean. De-
fault = 1.
increasing Sort in increasing order. Boolean. Default = 1.
decreasing Sort in decreasing order. Boolean. Default = 0.
duplicates Leave duplicated domain values in outpdm. If !duplicates
is specified then only the matrices from pdm1 associated
with the duplicate domain values are included in outpdm.
The value of tol determines what constitutes equality in
the domains. Boolean. Default = 1.
tol
Tolerance in determining duplicates in the domains. De-
fault = eps*max([domain(pdm1);domain(pdm2)])
Outputs:
outpdm
output pdm
Description
This function concatenates two pdms and then sorts them according to their domain.
This is useful for merging experimental data taken over different frequency or time
ranges. A regular domain is not required. The pdms must have equal row and column
dimensions.
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Example
time1 = [0:0.025:1]
pdm1 = gsin(time1,{frequency=2})
time2 = [0.8:0.02:1.5]
pdm2 = randpdm([],1,1,{dom=time2,zeromean})
outpdm = mergeseg(pdm1,pdm2)
gph1 = ctrlplot(outpdm);
gph1 = plot(pdm1,gph1,{marker=1,marker style=8,...
line=0});
gph1 = plot(pdm2,gph1,{marker=1,marker style=1,...
line=0,legend=["outpdm";"pdm1";"pdm2"]})?
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1
0.5
0
outpdm
pdm1
pdm2
-0.5
-1
0
0.5
1
1.5
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mkpert
Syntax
[pertsys] = mkpert(Delta,blk,mubnds,{fselect,pnorm,Hertz})
Parameter List
Inputs:
Delta
blk
Lower bound perturbation data from mu calculation.
(pdm)
block structure (refer to mu function documentation).
(matrix)
mubnds Calculated mu bounds (pdm).
Keywords: fselect
pnorm
Scalar valued. Specifies the frequency for interpolation,
overriding that obtained from mubnds.
Scalar valued. Specifies the norm of the resulting system,
overriding that calculated from mubnds.
Hertz
Boolean. Domain of pdms are in units of Hertz. This
keyword is mandatory. To specify radians/sec use !Hertz.
Outputs:
pertsys Constructed perturbation. (Dynamic System)
Description
Creates the rational, stable, system which interpolates the perturbation, Delta, at a
specified frequency. The frequency chosen is the one where the lower bound to mu is
maximum. This corresponds to the smallest destabilizing perturbation. The system has
the assumed structure given by blk.
This function may also be called with a single pdm argument. The result will interpolate
the argument at its first domain value.
This function is used to construct a bad perturbation, the effects of which can be
studied by simulation.
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The bounds, mubnds, are used to determine the ”worst-case” frequency for the
interpolation. The norm of pertsys is 1/(norm(Delta(jω))) where ω is the chosen
frequency. This is the smallest destabilizing perturbation at that frequency.
Both the interpolation frequency and the norm of pertsys can be specified via keywords.
The user may be interested only in certain frequency ranges and may have an assumed
norm bound on the perturbation (typically unity).
Components of pertsys (or all of it if appropriate) may be real valued gains rather than
dynamic systems.
Example
The use of mkpert is studied in context in the on-line help for musynfit and the manual
documentation for musynfit (page 299).
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mkphase
Syntax
[cdata] = mkphase(magdata, {skipchks,Hertz})
Parameter List
Inputs:
magdata Magnitude data (pdm)
Keywords: skipchks Boolean. Skip the error checking. (Default = 0)
Hertz
cdata
The domain of the pdm is in Hertz. This is the default.
!Hertz specifies a domain in rad/sec.
Outputs:
Complex valued data corresponding to a minimum phase
transfer function.
Description
Fits phase data to magnitude data. The phase is equivalent to that produced by
minimum phase system. A complex cepstrum method is used.
Reference
For further details see: “Digital Signal Processing,” A.V. Oppenheim & R.W. Schafer,
p.501, Prentice-Hall, 1975.
Example
The on-line help example is the same as that for fitsys. Refer to page 239.
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Limitations
Limited to SISO systems.
See Also
fitsys, ccepstrum
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modalstate
Syntax
outsys = modalstate(sys, {keywords})
Parameter List
Inputs:
sys
Input Dynamic System
Keywords: increasing Boolean. Order in terms of increasing magnitude (contin-
uous) or angle (discrete). Default = 1
decreasing Boolean. Order in terms of decreasing magnitude (contin-
uous) or angle (discrete). Default = 0
splitstable Boolean. Separate the stable from the unstable modes and
order separately. Default = 0.
Outputs:
outsys
output Dynamic System
Description
Transform the system to give a block diagonal A matrix, with complex eigenvalues in
2×2 blocks and real eigenvalues in 1×1 blocks. The system must be diagonalizable to
avoid introducing errors. The initial condition is also transformed. The state names are
deleted, however the input and output names, and period (if discrete) are preserved.
Example
sys = randsys(6,1,1,{stable})
rifd(sys)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
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-8.1602e-01
-8.1602e-01
-4.6142e+00
-4.6142e+00
-2.2478e+01
-3.9525e+01
1.3353e+00
-1.3353e+00
-1.6648e+01
1.6648e+01
0.0000e+00
0.0000e+00
1.5649e+00
1.5649e+00
1.7275e+01
1.7275e+01
2.2478e+01
3.9525e+01
0.5215
0.5215
0.2671
0.2671
1.0000
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0270e+00
-1.0270e+00
-4.5529e+00
-4.5529e+00
-2.4416e+01
-3.9534e+01
-1.5086e+00
1.5086e+00
-1.6621e+01
1.6621e+01
0.0000e+00
0.0000e+00
1.8250e+00
1.8250e+00
1.7233e+01
1.7233e+01
2.4416e+01
3.9534e+01
0.5628
0.5628
0.2642
0.2642
1.0000
1.0000
sys1 = modalstate(sys)
[a1,,,] = abcd(sys1)
# compare a1 to the poles of sys
a1?
a1 (a square matrix) =
-0.816024
1.33529
-1.33529
-0.816024
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-4.61417
-16.6478
16.6478
-4.61417
0
0
0
0
0
0
0
0
0
-22.4779
0
-39.5252
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mu
Syntax
[mubnds,D,Dinv,Delta,sens] = mu(M,blk)
Parameter List
Inputs:
M
Matrix or pdm.
blk
Block structure defined by a matrix of dimension: number
of blocks × 2. If the ith block has c outputs and r inputs,
then blk(i,:) = [r,c]. The default is equivalent to 1x1 blocks
(M must be square).
Outputs: mubnds Upper and lower bounds (in vector form) for mu(M).
D,Dinv D-scale matrices giving the calculated upper-bound.
mu(M) ≤ msv(D*M*Dinv)
Delta
sens
Perturbation achieving the lower bound.
Sensitivity of the upper bound with respect to the values
in D & Dinv
Description
Calculates the upper and lower bounds of the structured singular value of M, with block
structure: blk. The upper bound scaling matrices and the lower bound destabilizing
perturbation also returned.
The Osborne method is used to calculate the upper bound (for small matrices this is
enhanced by a Perron Frobenius method) and a power iteration is used for the lower
bound.
Example
# The following is the classic example showing
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# that mu is not equal to its upper bound for
# more than three full blocks.
gamma = 3 + sqrt(3); beta = sqrt(3) -1
a = sqrt(2/gamma); b = 1/sqrt(gamma)
c = 1/sqrt(gamma); d = -sqrt(beta/gamma)
f = (1+jay)*sqrt(1/(gamma*beta))
psi1 = -pi/2; psi2 = pi
U = [a,0; b,b; c,jay*c; d,f]
V = [0,a; b,-b; c,-jay*c; f*exp(jay*psi1), d*exp(jay*psi2)]
scl = diagonal(random(4,1)+0.1*ones(4,1))
M = scl*U*V*’*inv(scl)
# Consider 4 1x1 blocks
blk1 = [1,1; 1,1; 1,1; 1,1]
[mubnds1,D1,Dinv1,Delta1] = mu(M,blk1)
max(svd(M))?
ans (a scalar) =
2.81264
max(svd(D1*M*Dinv1))?
ans (a scalar) =
mubnds1?
1
mubnds1 (a column vector) =
1
0.860682
# Consider 1 4x4 block (equivalent to max sing. val.)
blk2 = [4,4]
[mubnds2,D2,Dinv2,Delta2] = mu(M,blk2)
# Note that the perturbation is such that
# det(I-M Delta) = 0.
max(svd(D2*M*Dinv2))?
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ans (a scalar) =
mubnds2?
2.81264
mubnds2 (a column vector) =
2.81264
2.81264
det(eye(4,4) - M*Delta2)?
ans (a scalar) =
-2.53156e-16 + 4.3828e-17 j
For an example of how mu is used for system robustness analysis, refer to the on-line
help for musynfit (page 299).
Limitations
This version of the software cannot handle repeated blocks or real valued blocks.
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musynfit
Syntax
[Dsys,Dinvsys] = musynfit(Dmag,blk,nmeas,nctrls,..
weight,M,order,{keywords})
Parameter List
Inputs: Dmag New D matrix from mu calculation (magnitude data only).
The domain is assumed to be in Hertz.
blk
nmeas scalar: number of measurements
nctrls scalar: number of controls
block structure (refer to mu function documentation).
weight (optional) weighting function for the fit. Typically the sen-
sitivity output of mu is used. Must be a pdm of dimension:
number of blocks × 1.
M
(optional) Matrix (pdm) of interest in the µ calculation.
I.e. µ(M) ≤ σ(DMD−1). If M is provided a second plot
compares the µ(M) upper bound, based on Dmag, to the
mu(M) upper bound using a frequency response of the D
scale transfer function approximation.
order
(optional) A scalar (or vector of dimension nblks-1) spec-
ifying the order of the fit to be used. If this argument is
present, the function runs without requiring user interac-
tion and does not graph the results.
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Keywords: Hertz
Boolean. This keyword is mandatory as the function must
know whether the domain is in Hertz or radians/second
(specified by !Hertz) to fit correctly. Note that the Xmath
function freq assumes that the frequency range is specified
in Hertz.
plotweight Boolean, default = 0. This will generate a second plot
showing the weighting function. This can be useful is as-
sessing the quality of a given transfer function fit. If the
variable M is also specified, the weighting function plot
takes priority over the upper bound comparison plot.
fit
Integer specifying the routine to be used in fitting the
data. This sets the default routine which may be changed
interactively by the user. The choices are:
fit = 1
fit = 2
Xmath function: tfid
Xmu function: fitsys
The default is fit = 2.
Outputs:
Dsys
New left D scale system.
Dinvsys
New D inverse system (to be multiplied to M on the right).
Description
Fits stable, minimum phase, transfer functions to each block of the D-scale matrix. A
complex cepstrum technique is used to generate the phase response from the provided
magnitude data.
The user may interactively select a fitting order and compare the result with the data. A
choice of fitting routines is provided. If the order is specified in the command line the
function is not interactive.
Example
# The nominal plant is a double integrator.
# A multiplicative perturbation weight reflects
# increased uncertainty at high frequencies
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P = 1/makepoly([1,0,-0.01],"s")
W = makepoly([1,20],"s")/makepoly([1,200],"s")
# Set up an unweighted interconnection structure
# for unity gain negative feedback. We include
# the perturbation too.
nms = ["P";"W"]
inp = ["delt";"ref";"noise";"control"]
outp = ["W" ;"ref- delt - P";"control";"ref-delt-P-noise"]
cnx = ["control";"P"]
ic = sysic(nms,inp,outp,cnx,P,W)
# Now include some weights for performance:
Wperf = makepoly([0.01,1],"s")/makepoly([1,0.01],"s")
Wact = 0.1* makepoly([1,1],"s")/makepoly([0.05,1],"s")
Wnoise = 0.01
Wref = makepoly([0.005,1],"s")/makepoly([0.05,1],"s")
# Apply weights to error signals and unknown
# inputs.
wghtic = daug(1,Wperf,Wact,1)*ic*daug(1,Wref,Wnoise,1)
# Perform an H infinity design.
nmeas = 1
ncntrls = 1
gmin = 0
# number of measurements
# number of controls
gmax = 10
Kinf = hinfsyn(wghtic,nmeas,ncntrls,[gmin;gmax])
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
10.0000
Y eig
Hx eig
X eig
nrho xy
0.0049
0.0207
0.1061
2.3572
0.2681
p/f
10.000
5.000
2.500
1.250
1.875
7.1e-01
7.1e-01
7.0e-01
7.7e-04
7.7e-04
7.8e-04
9.9e-03 -3.8e-19
9.8e-03 -1.5e-19
9.2e-03
6.0e-03
p
p
p
f
p
0.0e+00
0.0e+00
6.5e-01 -5.6e+02
6.9e-01 7.9e-04
8.5e-03 -1.1e-18
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1.562
1.406
1.484
1.523
6.8e-01
6.7e-01
6.7e-01
6.7e-01
8.0e-04
8.1e-04
8.0e-04
8.0e-04
7.7e-03
7.0e-03 -1.1e-18
7.4e-03 0.0e+00
7.5e-03 -5.0e-19
0.0e+00
0.6811
2.0900
1.0394
0.8249
p
f
f
p
Gamma value achieved:
G = starp(wghtic,Kinf)
1.5234
omega = logspace(0.01,100,25)
Gg = freq(G,omega)
G11g = Gg(1,1)
G22g = Gg(2:3,2:3)
rs = max(svd(G11g))
np = max(svd(G22g))
blk = [1,1; 2,2]
[rpbnds1,D1,Dinv1,Delta1,sens1] = mu(Gg,blk)
gph1 = ctrlplot([np;rs;rpbnds1],{log});
gph1 = plot(gph1,{!grid,title="mu analysis",legend=["nominal
perf.";...
"robust stab.";"robust perf.(upper)";"robust perf.
(lower)"]})?
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mu analysis
1.2
1
0.8
0.6
0.4
0.2
0
nominal perf.
robust stab.
robust perf.(upper)
robust perf. (lower)
0.01
0.1
1
10
100
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# Fit transfer functions to D1 & Dinv1 for a mu
# synthesis iteration
[Ds,Dinvs] = musynfit(D1,blk,nmeas,ncntrls,sens1,Gg,{Hertz})
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D scale fit, block: 1
10000
1000
100
10
Magnitude data
Previous fit, order: 0
New fit, order: 3
1
0.01
0.1
1
10
10
10
100
100
100
Frequency (Hz)
Upper bound comparison, block: 1
100
10
1
Data based bound
0.1
Previous fit bound, order: 0
New fit bound, order: 3
0.01
0.001
0.01
0.1
1
Frequency (Hz)
D scale fit weight for block: 1
1
0.1
0.01
0.001
0.0001
1e-05
0.01
0.1
1
Frequency (Hz)
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# Apply the D scales to another H infinity design
Kmu = hinfsyn(Ds*wghtic*Dinvs,nmeas,ncntrls,[gmin;gmax])
Test bounds:
gamma
0.0000
<
gamma <=
Hy eig
10.0000
Y eig
Hx eig
X eig
nrho xy
0.0027
0.0113
0.0517
0.4688
*******
*******
1.2451
0.6922
p/f
10.000
5.000
2.500
1.250
0.625
0.938
1.094
1.172
6.5e-01
6.4e-01
6.4e-01
6.3e-01
5.6e-07
5.6e-07
5.6e-07
5.6e-07
9.9e-03 -1.0e-15
9.8e-03 -6.3e-16
9.2e-03
6.0e-03 -7.6e-20
7.1e-14 *******
7.1e-14 *******
4.1e-03 -3.5e-16
5.2e-03 -6.3e-18
p
p
p
p
0.0e+00
5.6e-01 -1.1e+00
6.1e-01 -2.8e+01
f
f
6.2e-01
6.2e-01
5.6e-07
5.6e-07
f
p
Gamma value achieved:
1.1719
# Close the loop around the weighted interconnection
# structure.
Gmu = starp(wghtic,Kmu)
omega = logspace(0.01,100,40)
Gmug = freq(Gmu,omega)
blk = [1,1; 2,2]
[rpbnds2,D2,Dinv2,Delta2,sens2] = mu(Gmug,blk)
gph3 = ctrlplot(rpbnds2(1,1),{log});
gph3 = ctrlplot(rpbnds1(1,1),gph3,{log});
gph3 = plot(gph3,{!grid,title="Kmu & Kinf mu analysis",...
legend=["Kmu: robust perf.";"Kinf: robust perf."]})?
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Kmu & Kinf mu analysis
1.2
1
0.8
0.6
0.4
0.2
0
Kmu: robust perf.
Kinf: robust perf.
0.01
0.1
1
10
100
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# Look at the worst case perturbations for each of
# the Kinf and Kmu controllers. Compare also a
# random perturbation for each controller. In all cases the
# perturbation is of size 0.5 and we choose a perturbation
# which is bad at 1Hz.
mupert = mkpert(Delta2,blk,rpbnds2,{fselect=1,pnorm=0.5,Hertz})
mupert = mupert(1,1)
# select part to close around top
infpert = mkpert(Delta1,blk,rpbnds1,{fselect=1,pnorm=0.5,Hertz})
infpert = infpert(1,1) # select part to close around top
# set up an LFT for the reference tracking problem
M = daug(W,1,1)*consys([0,0,1;1,0,1;-1,1,-1])*daug(1,1,P)
mupertic = starp(mupert,M)
mupertclp = starp(mupertic,Kmu)
munomclp = starp(starp(0,M),Kmu)
infpertic = starp(infpert,M)
infpertclp = starp(infpertic,Kinf)
infnomclp = starp(starp(0,M),Kinf)
rpert = randpert(blk,{pnorm=0.5})
rpert = rpert(1,1)
rpertic = starp(rpert,M)
murpert = starp(rpertic,Kmu)
infrpert = starp(rpertic,Kinf)
# Look at step responses
time = [0:0.05:4]
step = gstep(time,0,1)
yinfnom = infnomclp*step
yinfpert = infpertclp*step
ymunom = munomclp*step
ymupert = mupertclp*step
gph4 = ctrlplot([step,yinfnom(1,1),yinfpert(1,1),...
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ymunom(1,1),ymupert(1,1)]);
gph4 = plot(gph4,{legend=["input step";"Kinf nominal";...
"Kinf pert.";"Kmu nominal";"Kmu pert."]})?
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1.4
1.2
1
0.8
0.6
0.4
0.2
0
input step
Kinf nominal
Kinf pert.
Kmu nominal
Kmu pert.
0
1
2
3
4
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# Compare with a random perturbation
yinfrandp = infrpert*step
ymurandp = murpert*step
gph5 = ctrlplot([step,yinfnom(1,1),yinfrandp(1,1),...
ymunom(1,1),ymurandp(1,1)]);
gph5 = plot(gph5,{legend=["input step";"Kinf nominal";...
"Kinf rand. pert.";"Kmu nominal";"Kmu rand. pert."]})?
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1.4
1.2
1
0.8
0.6
0.4
0.2
0
input step
Kinf nominal
Kinf rand. pert.
Kmu nominal
Kmu rand. pert.
0
1
2
3
4
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See Also:
mu, hinfsyn, mkpert, hinfnorm.
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ophank
Syntax
[SysR,SysU,HSV] = ophank(Sys,{nsr,onepass})
Parameter List
Inputs:
Sys
nsr
Linear, stable, state-space system (continuous)
(optional) Order of the reduced system. If not specified,
the user will be prompted for its value after the Hankel
singular values are displayed.
Keywords: onepass (Boolean) If equal to 1 (true), reduction is calculated in
one pass. If false (!onepass or onepass=0), reduction is
calculated in (number of states of Sys - nsr) passes. De-
faults to 1.
Outputs:
SysR
SysU
Reduced order system; dynamic system object.
Anti-causal optimal system, (only with one keyword); dy-
namic system object.
HSV
A column vector containing the Hankel singular values of
the system, Sys.
Description
Calculates an optimal hankel norm reduction of Sys for the additive case.
Any initial state values or state names associated with Sys are not assigned to SysR or
SysU. Input and Output names are maintained.
The current version of ophank is unable to deal with discrete time systems. Users are
advised to call makecontinuous on discrete systems before calling ophank, and then
re-discretize.
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Uses additional subroutines ophiter, ophred, ophmult and stable.
This function is cross-licensed from the Model Reduction Module.
Example
# Create a five state system for reduction.
a = daug(-0.891334,[-1.20857,0.799042;-0.799042,-1.20857],...
-4.74685,-21.3013)
b = [0.0262569;-0.189601;-0.113729;0.211465;-0.538239]
c = [0.120725,-0.336942,0.397198,-0.700524,-1.02235]
d = 0
sys1 = system(a,b,c,d)
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
# Reduce to 3 states by Hankel norm approximation
[sysout1,hsv] = ophank(sys1,{nsr=3})
sysout1g = freq(sysout1,fHz)
balerr = sys1g - sysout1g
# Reduce to a 3 state system by residualization
# for comparison purposes.
sysout2 = sresidualize(sys1,3)
sysout2g = freq(sysout2,fHz)
residerror = sys1g - sysout2g
gph1 = ctrlplot([sys1g,sysout1g,sysout2g,...
balerr,residerror],{logmagplot});
gph1 = plot(gph1,{!grid,legend=["original system";...
"reduced: ophank";"reduced: sresidualize";...
"ophank error";"sresidualize error"]})?
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0.1
0.01
0.001
original system
reduced: ophank
reduced: sresidualize
ophank error
sresidualize error
0.0001
0.01
0.1
1
10
100
Frequency
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See Also
minimal, balmoore.
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orderstate
Syntax
outsys = orderstate(sys,indx)
Parameter List
Inputs:
sys
Input Dynamic System
indx
Lists the desired order of the states in outsys.
Outputs: outsys output dynamic system.
Description
Reorder the states according to the index argument. The state names and initial
condition are also ordered. The input and output names, and period (if discrete) are
preserved.
Example
sys1 = randsys(3,1,1,{stable})
sys1 = system(sys1,{statenames=["s1";"s2";"s3"],...
x0=[0.1;0.2;0.3]})?
sys1 (a state space system) =
A
-2.27378
0.65518
-1.2488
3.32295
-5.41526
4.7424
-8.82005
9.0643
-13.2781
B
0.0879738
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0.710595
0.688873
C
0.659532
0.181512
0.390497
D
0.15869
X0
0.1
0.2
0.3
State Names
-----------
s1
s2
s3
System is continuous
sys2 = orderstate(sys1,[2,1,3])?
sys2 (a state space system) =
A
-5.41526
3.32295
4.7424
0.65518
-2.27378
-1.2488
9.0643
-8.82005
-13.2781
B
0.710595
0.0879738
0.688873
C
0.181512
0.659532
0.390497
D
0.15869
X0
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0.2
0.1
0.3
State Names
-----------
s2
s1
s3
Input Names
-----------
Input 1
Output Names
------------
Output 1
System is continuous
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randpdm
Syntax
pdmout = randpdm (ndomain,nrows,ncolumns,{keywords})
Parameter List
Inputs:
ndomain
nrows
length of the domain
number of rows in pdmout
ncolumns number of columns in pdmout
Keywords: complex
Boolean. A complex valued pdm is generated. Default =
0.
zeromean Boolean. The values are shifted so that zero is the mean.
Default = 0.
Dfirst
First value in the domain. This only a bound if !regular.
Default = 0.
Last value in the domain. Again this is only a bound if
!regular. Default = ndomain-1.
Dlast
regular
Boolean. Domain is regular. !regular generates a random
domain between Dfirst and Dlast. Default = 1.
Outputs:
pdmout
Random pdm
Description
A random pdm, with user specified row, column and domain dimension, is generated.
Several additional features can be specified by keywords: real or complex values,
minimum and maximum values of the domain, and whether or not the domain is regular.
Note that the domain related defaults give a domain of [0:ndomain:1]. ndomain can be
specified as zero in which case a random matrix is returned. This is a reasonable way of
generating a random complex matrix.
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Examples
pdm0 = randpdm(3,1,2,{zeromean})?
pdm0 (a pdm) =
domain |
-------+-------------------------
0 | -0.043917 -0.0789571
-------+-------------------------
1 | 0.37878 -0.457265
-------+-------------------------
2 | 0.0466939 -0.34413
Col 1
Col 2
-------+-------------------------
pdm1 = randpdm(4,1,1,{!regular,Dfirst=2,Dlast=19})?
pdm1 (a pdm) =
domain |
---------+------------
3.1543
---------+------------
4.87235 | 0.900169
---------+------------
10.0323 0.932966
---------+------------
18.3191 0.645692
|
0.0605523
|
|
---------+------------
pdm2 = randpdm(pdm1)?
pdm2 (a pdm) =
domain |
---------+------------
3.1543
---------+------------
4.87235 | 0.174411
---------+------------
|
0.746793
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10.0323
---------+------------
18.3191 0.81113
|
0.922528
|
---------+------------
pdm3 = randpdm(0,3,2,{complex,zeromean})?
pdm3 (a rectangular matrix) =
-0.237052 - 0.10845
0.097563 + 0.0140395 j
0.0178566 - 0.153032
j
-0.250958 - 0.392096 j
-0.255142 - 0.419362 j
0.0986898 + 0.375528 j
j
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randpert
Syntax
[pert] = randpert(blk, {sys,sfreq,complex,pnorm})
Parameter List
Inputs:
blk
block structure (refer to mu function documentation).
(matrix)
Keywords: sys
sfreq
Boolean. Specifies that pert is a dynamic system. Default
= !sys, i.e. pert is a matrix.
Scalar valued. If a dynamic system is specified, sfreq is the
frequency with significant phase. Units are Hertz. Default
= 1.0.
complex Boolean. Specifies that pert is complex valued. Default =
1.
pnorm
Scalar valued. Specifies the norm of the resulting system.
Default = 1.
Outputs:
pert
Constructed perturbation
Description
Creates a perturbation of the structure specified by blk. By default this is a complex
valued matrix. The sys keyword will create an allpass Dynamic System. sfreq is used
to specify a frequency (in Hertz) where there is significant phase in the system.
This function is used to construct random perturbations, the effects of which can be
studied by simulation. Dynamic perturbations are appropriate for simulating the effects
of unmodeled dynamics. In such cases, the crossover frequency, or a frequency where mu
is large, are good choices.
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Example
The use of randpert is studied in context in the on-line help for musynfit and the
manual documentation for musynfit (page 299).
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randsys
Syntax
sys = randsys (nstates,noutputs,ninputs,{keywords})
Parameter List
Inputs:
nstates
number of states in sys
number of outputs in sys
number of inputs in sys
noutputs
ninputs
Keywords: stable
Boolean. sys is forced to be stable (default = 1)
oscillatory Boolean. Oscillatory poles are allowed. (default = 1). !os-
cillatory gives only real eigenvalues (continuous) or positive
real eigenvalues (discrete).
discrete
dt
Boolean. Generate a discrete time system. (default = 0).
Sample period for discrete time system (default =
1/(2*Fmax)).
Fmax
Fmin
upper bound of sys pole frequencies (Hz) (default = 10 Hz
or = 1/(2*dt) if dt specified)
lower bound of sys poles frequencies (Hz) (default =
Fmax/100 Hz)
Dterm
Select a random D term (default = 1: a D term is choosen)
linear dist linear distribution of poles. The default is uniform (or
normal) distribution on a log frequency axis. A linear fre-
quency axis distribution may be more representative of
some systems.
Outputs:
sys
Random Dynamic System
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Description
A random system, with user specified state, input and output dimension, is generated.
Several additional features can be specified by keywords: whether or not the system is
stable, where or not it can contain oscillatory modes, whether or not it has a D term,
and bounds on the minimum and maximum pole frequencies.
If the discrete keyword is specified, a bilinear transformation is performed to get the
random discrete system. A sample time can be specified with dt.
Examples
sys1 = randsys(4,1,2,{stable,!oscillatory,...
Fmin=0.1,Fmax=10})
size(sys1)?
ans (a row vector) =
rifd(sys1)?
1
2
4
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.7920e+00
-5.2688e+00
-5.6953e+00
-5.3738e+01
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
2.7920e+00
5.2688e+00
5.6953e+00
5.3738e+01
1.0000
1.0000
1.0000
1.0000
Zeros:
sys2= randsys(10,1,1,{stable,oscillatory,Fmin=0.1,...
Fmax=1,!Dterm})
rifd(sys2)?
Poles:
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real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0408e+00
-1.1390e+00
-2.8050e-01
-2.8050e-01
-1.5337e+00
-5.5566e-01
-5.5566e-01
-3.1454e+00
-5.6798e+00
-6.1039e+00
0.0000e+00
0.0000e+00
-1.4858e+00
1.4858e+00
0.0000e+00
-1.6958e+00
1.6958e+00
0.0000e+00
0.0000e+00
0.0000e+00
1.0408e+00
1.1390e+00
1.5120e+00
1.5120e+00
1.5337e+00
1.7845e+00
1.7845e+00
3.1454e+00
5.6798e+00
6.1039e+00
1.0000
1.0000
0.1855
0.1855
1.0000
0.3114
0.3114
1.0000
1.0000
1.0000
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-8.6788e-02
-8.6788e-02
-1.0395e+00
-1.5152e+00
-6.3796e-01
-6.3796e-01
-3.1425e+00
-5.8916e+00
-5.8916e+00
-1.0161e+00
1.0161e+00
0.0000e+00
0.0000e+00
-1.9633e+00
1.9633e+00
0.0000e+00
1.5472e-01
-1.5472e-01
1.0198e+00
1.0198e+00
1.0395e+00
1.5152e+00
2.0643e+00
2.0643e+00
3.1425e+00
5.8937e+00
5.8937e+00
0.0851
0.0851
1.0000
1.0000
0.3090
0.3090
1.0000
0.9997
0.9997
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
sys2g = freq(sys2,fHz)
gph1 = ctrlplot([sys1g,sys2g],{bode});
gph1 = plot(gph1,{legend=["sys1 (input 1)";...
"sys1 (input 2)";"sys2"]})?
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10
1
0.1
sys1 (input 1)
sys1 (input 2)
sys2
0.01
0.001
0.01
0.1
1
10
100
Frequency
20
0
-20
-40
-60
-80
-100
0.01
0.1
1
10
100
Frequency
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sys3 = randsys(6,1,1,{discrete,dt=5})
rifd(sys3)?
Poles:
radius
angle
(radians)
0.1652
0.0000
0.0000
-0.0394
0.0394
-0.2163
0.2163
0.4580
0.7872
0.7872
0.9307
0.9307
Zeros:
radius
angle
(radians)
0.1623
0.7981
0.6658
0.8921
0.8921
15.4278
0.0000
0.0000
0.0000
-0.1981
0.1981
3.1416
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rifd
Syntax
[stat] = rifd(vec,{discrete,Hertz,degrees})
Parameter List
Inputs:
vec
complex valued vector (or Dynamic System - see below).
Keywords: discrete Boolean. Vector is to be interpreted in the z domain,
rather than the s domain. Default = 0.
Hertz
Boolean. Display frequency units in Hertz on the s plane.
!Hertz gives a display in radians/sec. Default = 0.
degrees Boolean. Display angle in degrees for z plane. !degrees
gives a display in radians. Default = 0.
Outputs:
stat
Status. stat = 1 if an error occurs.
Description
Displays complex valued vectors in terms of s or z domain properties. The primary use
is for interpreting the output of poles or zeros in engineering units.
For the s domain, the real and imaginary parts and frequency and damping are
displayed.
For the z domain, the pole radius and angle are displayed. Note that the angle is
actually the normalized frequency (radians/sample).
If vec is a Dynamic System, the poles and zeros are calculated and displayed. This is
a useful shorthand for the most common usage: rifd(poles(system)). In this case
there is no need to specify the discrete keyword as this is be determined directly from
the Dynamic System.
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Examples
sys1 = randsys(4,3,2,{stable})
rifd(sys1)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.6847e+00
-2.4383e+00
-8.7457e+00
-1.5041e+01
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
1.6847e+00
2.4383e+00
8.7457e+00
1.5041e+01
1.0000
1.0000
1.0000
1.0000
Zeros:
ans (a scalar) =
0
# Compare rifd to A matrix eigenvalues
[a,b,c,d] = abcd(sys1)
Aeigs = eig(a)?
Aeigs (a column vector) =
-1.68474
-2.43829
-8.7457
-15.0408
rifd(Aeigs)
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.6847e+00
-2.4383e+00
-8.7457e+00
-1.5041e+01
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
1.6847e+00
2.4383e+00
8.7457e+00
1.5041e+01
1.0000
1.0000
1.0000
1.0000
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sys2 = randsys(3,3,2,{stable,discrete})
rifd(sys2)
Poles:
radius
angle
(radians)
0.9641
0.2985
0.2985
0.0000
-0.5348
0.5348
Zeros:
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sdtrsp
Syntax
[v,y,u] = sdtrsp(Sys,dSys,w,tfinal,...
{ord,intstep,cdelay})
Parameter List
Inputs:
Sys
Continuous dynamic system. This is the upper system in
the LFT. The initial states are used in the simulation.
Digital dynamic system. Lower system in the LFT. The
initial states are used in the simulation.
dSys
w
PDM. Input signal.
tfinal
Final time in the simulation (optional). Default = max
time specified in w.
Keywords: ord
Scalar valued. Specifies order of interpolation for continu-
ous signals in the system. Options are 0 or 1. Default =
0.
intstep Scalar valued. Integration step size. If not supplied a
default will be calculated based on the system eigenvalues.
It will be rounded to make it divide into the discrete system
sample time by an integer value of at least 2.
cdelay Scalar value. Delay implemented at the output to dSys.
(in sec.) Optional, default = 0. It must be less than the
digital system sample time.
Outputs:
v
y
u
Signal from upper system in the LFT.
Signal into lower system in the LFT.
Signal out of the lower system in the LFT.
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Description
Time domain simulation of a sampled data interconnection. The applicable closed loop
system is illustrated below.
v
w
Sys
y
u
-
dSys
This is conceptually the equivalent of:
v = starp(Sys, dSys) ∗ w.
This function will handle interconnections in which the continuous time signals are not
necessarily synchronized with the digital system. Computation delays which are a
fraction of the sample period may also be simulated. The following equations represent
the discrete system,
z(kT + T )
=
=
Adz(kT ) + Bdy(kT )
u(kT + cdelay)
Cdz(kT ) + Ddy(kT )
For delays of greater than a sample period append additional states to dSys to model the
integer sample period part of the delay. Use cdelay to model the fractional remainder.
The actual calculation is performed with a fast discrete time equivalent. Intstep is the
fast integration step size. The input vector, w will be interpolated to the same step size.
Either a zero or first order interpolation is used, depending on the keyword ord. If ord =
0 a zero-order hold equivalent is used for the continuous plant. In the ord = 1 case a
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triangle hold equivalent is used. This is the same a linearly connecting the samples at
the input to the hold.
Some care is needed in the choice of cdelay and intstep. The default for intstep is based
on the continuous system eigenvalues and the minimum time spacing in the input vector,
w. It will be forced to be an integer divisor of the digital sample time. Intstep must also
be a divisor of cdelay. The user can inadvertently force a very small integration period
by selecting cdelay without regard to the sample period. Warnings are printed if this is
suspected to be the case.
The upper LTF block (continuous sytem) can be a constant matrix. The lower block
(digital system) must be a discrete dynamic system in order to specify the sampling
time. If a constant gain digital system is required specify it with,
dSys = system([], [], [], Dd, T).
Example:
# Consider a continuous time plant,
P = 1/makepoly([1,0,-0.05],"s")
[a,b,c,d] = abcd(P)
P = system(a,b,c,d)
T = 1/20
# sample period
# The following digital controller is used.
# The first input is the reference and the # second is the
measurement.
digC = system([-1.5,T/4; -2/T,-.5],[.5,2;1/T,1/T],...
[-1/T^2,-1.5/T], [1/T^2,0],T)
# Now consider a sample & hold version of P and
# check that the closed loop digital system is stable
# and has a reasonable step response.
digP = discretize(P,T,exponential)
snm = ["digP"; "digC"]
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inps = "ref"
outs = "digP"
cnx = ["digC"; ...
"[ ref;digP]"]
# input to digP
# input to digC
digclp = sysic(snm,inps,outs,cnx,digP,digC)
# Calculate the digital system step response
time = [0:T:20*T]
step = gstep(time,0,1)
digy = digclp*step
# Do a complete sampled data simulation. We
# will assume that the controller has an input to
# output calculation delay of 0.1*T. The
# plant will also have a non-zero initial
# condition and a sinusoidal disturbance will
# be imposed at the output. The disturbance has
# deliberately been chosen at above the Nyquist
# frequency.
#P = system(P,x0=[0.25;0])
calcdelay = 0.1*T
# initial condition
# calculation delay
finetime = [0:0.005:1]
dist=0.25*gsin(finetime,frequency=11) # disturbance
step = gstep(finetime,0,1) # step
# calculate the continuous time part of the
# system as the upper LFT object.
ctssys = consys([0,1,1;1,0,0;0,1,1])*daug(1,1,P)
ctssys = system(ctssys,x0=[0.25;0])
ctsy = sdtrsp(ctssys,digC,[step;dist],...
cdelay=calcdelay)
gph1 = ctrlplot(ctsy);
gph1 = plot(gph1,digy,{line=0,marker=1,...
legend= ["sampled data calc.";"discrete calc."]})?
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4
3
sampled data calc.
discrete calc.
2
1
0
-1
-2
0
0.2
0.4
0.6
0.8
1
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See Also
trsp
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simtransform
Syntax
out = simtransform(sys,X)
Parameter List
Inputs:
sys Input system. This may be a state-space system, pdm or
constant matrix.
X
Similarity transform. X must be invertible
Outputs: out output system - in the same class as the input.
Description
Apply a similarity transform to sys. If sys is a matrix or pdm this gives out =
inv(X)*sys*X. If sys is a Dynamic System this transform is applied to the state.
Applying this transform to a transfer function is meaningless as a state matrix has not
been uniquely defined. This will return a warning and leave the transfer function
unchanged.
This command will remove the state names as they no longer have any meaning.
Example
sys1 = randsys(3,1,1,{stable})
rifd(sys1)
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
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-1.9737e+01
-9.7569e+00
-9.7569e+00
0.0000e+00
-2.9129e+01
2.9129e+01
1.9737e+01
3.0720e+01
3.0720e+01
1.0000
0.3176
0.3176
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-2.1990e+01
-9.7962e+00
-9.7962e+00
0.0000e+00
-2.9186e+01
2.9186e+01
2.1990e+01
3.0786e+01
3.0786e+01
1.0000
0.3182
0.3182
[a,b,c,d] = abcd(sys1)
# Do a schur transform on the A matrix
# and use for a similarity transform on the system.
[as,X] = schur(a,{real})
sysout = simtransform(sys1,X)?
sysout (a state space system) =
A
-19.737
3.66249
-9.75694
-17.1748
4.10895
49.404
-9.75694
0
-8.88178e-16
B
1.11535
0.042032
-0.151334
C
1.17657
-0.369351
-0.163514
D
0.566721
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X0
0
0
0
Input Names
-----------
Input 1
Output Names
------------
Output 1
System is continuous
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spectrad
Syntax
out = spectrad(mat)
Parameter List
Inputs:
mat Square matrix or pdm.
Outputs: out spectral radius of the input.
Description
Calculates the spectral radius (magnitude of the maximum eigenvalue) of the input
matrix.
Example
pdm1 = randpdm(3,2,2,{zeromean})?
pdm1 (a pdm) =
domain |
Col 1
Col 2
-------+------------------------------
0 | Row 1
| Row 2
0.0504605 -0.0914956
0.221744 -0.0231464
-------+------------------------------
1 | Row 1
0.139306
0.496387
| Row 2 -0.342521
0.0350694
-------+------------------------------
2 | Row 1 -0.287094 0.0591451
| Row 2 -0.0695034 -0.477195
-------+------------------------------
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eig(pdm1)?
ans (a pdm) =
domain |
-------+-------------------------------
0 | Row 1
| Row 2
0.013657 + 0.137601 j
0.013657 - 0.137601 j
-------+-------------------------------
1 | Row 1
| Row 2
0.0871876 + 0.409031 j
0.0871876 - 0.409031 j
-------+-------------------------------
2 | Row 1 -0.311974
| Row 2 -0.452314
-------+-------------------------------
spectrad(pdm1)?
ans (a pdm) =
domain |
-------+-----------
0 |
-------+-----------
1 | 0.41822
-------+-----------
2 | 0.452314
-------+-----------
0.138277
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sresidualize
Syntax
sysout = sresidualize(sysin,ord)
Parameter List
Inputs:
sysin
ord
Input Dynamic System
Order of sysout
Outputs: sysout output Dynamic System
Description
Residualize the states of sysin to the number specified by ord. The last nx−ord states
(nx is the number of states in sysin) are residualized. If ord is greater than the number
of states in sysin then sysout = sysin and a warning is displayed. sysin may be a
constant matrix, in which case it is treated as a system with zero states.
Example
# Create a five state system for reduction.
a = daug(-0.891334,[-1.20857,0.799042;-0.799042,-1.20857],...
-4.74685,-21.3013)
b = [0.0262569;-0.189601;-0.113729;0.211465;-0.538239]
c = [0.120725,-0.336942,0.397198,-0.700524,-1.02235]
d = 0
sys1 = system(a,b,c,d)
# Reduce to a 3 state system by residualization
# and truncation.
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sysout1 = sresidualize(sys1,3)
sysout2 = truncate(sys1,3)
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
sysout1g = freq(sysout1,fHz)
sysout2g = freq(sysout2,fHz)
residerror = sys1g - sysout1g
truncerror = sys1g - sysout2g
gph1 = ctrlplot([sys1g,sysout1g,sysout2g,residerror,...
truncerror],{logmagplot});
gph1 = plot(gph1,{!grid,legend=["original system";...
"residualized system";"truncated system";...
"residualization error";"truncation error"]})?
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0.1
0.01
0.001
0.0001
1e-05
original system
residualized system
truncated system
residualization error
truncation error
0.01
0.1
1
10
100
Frequency
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See also
rifd, simtransform, orderstate modalstate, truncate.
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starp
Syntax
out = starp (upper,lower,dim1,dim2,skipChks)
Parameter List
Inputs:
upper
Upper object (Dynamic System, constant or pdm) in the
interconnection
lower
dim1
Lower object in the interconnection.
The number of outputs of upper to be connected as inputs
to lower. (Optional - see description for default)
dim2
The number of outputs of lower to be connected as inputs
to upper. (Optional - see description for default)
Keywords: skipChks (Boolean, default = 0). Don’t check input arguments – this
includes type (Dynamic System, pdm) consistency and
fractional well posedness. For pdms, the well posedness
test may involve significant computation.
Outputs:
out
Resulting interconnection.
Description
Connects two objects in a Redheffer “star product,” as illustrated in the following
diagram.
This results in the following system.
Constant matrices can be interconnected with either pdms or dynamical systems. As
expected a Dynamic System and a pdm cannot be interconnected in this way.
If dim1 and dim2 are omitted, it is assumed that a linear fractional transformation is
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specified. This is equivalent to:
dim1 = min(upper output dim, lower input dim)
and
dim2 = min(upper input dim, lower output dim).
Examples
# Look at a constant matrix,
M = random(4,4)
lower = [3,4]
# The interconnected LFT will have 2 rows and 3 columns
result = starp(M,lower)?
result (a rectangular matrix) =
0.261939
-0.648616
-0.558202
-1.11273
0.484644
0.107883
# This is equivalent to:
M11 = M(1:2,1:3)
M12 = M(1:2,4)
M21 = M(3:4,1:3)
M22 = M(3:4,4)
M11 + M12*lower*inv(eye(2,2)-M22*lower)*M21?
ans (a rectangular matrix) =
0.261939
-0.648616
-0.558202
-1.11273
0.484644
0.107883
# A much more common usage is to provide an interconnection
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# structure for closing control loops. The following is
# the structure for a simple unity gain negative feedback
# system with plant P.
P = 1/makepoly([1,1],"s")
M = consys([0,1;1,-1])*daug(1,P)
# Test this with a Proportional and PI controller
Kp = 10
KpInt = makepoly([10,100],"s")/makepoly([1,0],"s")
rifd(starp(M,Kp))
Poles:
# look at poles and zeros
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.1000e+01
Zeros:
0.0000e+00
1.1000e+01
1.0000
rifd(starp(M,KpInt))
Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.5000e+00
-5.5000e+00
-8.3516e+00
8.3516e+00
1.0000e+01
1.0000e+01
0.5500
0.5500
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
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-1.0000e+01
0.0000e+00
1.0000e+01
1.0000
# Now consider the system with an output multiplicative
# perturbation (of 10%)
W = consys(0.1)
G = daug(W,1,1)*consys([0,0,1;1,0,1;-1,1,-1])*daug(1,1,P)
# The nominal system is constructed by closing the
# upper loop with 0. Any thing else is a perturbed
# system.
Gnom = starp(0,G)
Gpert = starp(-1,G)
# Now examine the closed loop affects for the PI
# controller.
rifd(starp(Gnom,KpInt))
Poles:
# same as before
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.5000e+00
-5.5000e+00
-8.3516e+00
8.3516e+00
1.0000e+01
1.0000e+01
0.5500
0.5500
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0000e+01
0.0000e+00
1.0000e+01
1.0000
rifd(starp(Gpert,KpInt))
# perturbation moves the poles.
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Poles:
real
imaginary
frequency
(rad/sec)
damping
ratio
-5.0000e+00
-5.0000e+00
-8.0623e+00
8.0623e+00
9.4868e+00
9.4868e+00
0.5270
0.5270
Zeros:
real
imaginary
frequency
(rad/sec)
damping
ratio
-1.0000e+01
0.0000e+00
1.0000e+01
1.0000
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substr
Syntax
littlestring = substr(bigstring,charindex,{skipChks})
Parameter List
Inputs:
bigstring
Input string (a 1×1 string matrix)
vector indexing the characters to be returned in the output.
charindex
Keywords: skipChks
Boolean specifying that syntax checking is to be skipped.
Outputs:
littlestring Output string
Description
substr takes characters from the string, bigstring and concatentates them to from the
output, littlestring. The vector, charindex, determines the characters and their
order.
This function allows the user to select substrings from an input string. However, as
shown in the example, there is no requirement that the characters specified in charindex
be contiguous or non-repeated.
To find a specified character in a string use the function: index.
Example
alphabet = " abcdefghijklmnopqrstuvwxyz"
mantra = substr(alphabet,[14,22,1,19,22,13,6,20])?
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sysic
Syntax
[sys] = sysic(sysNames,sysInputs,sysOutputs,sysConnects,...
subsys1,subsys2,...)
Parameter List
Inputs:
sysNames
sysInputs
A vector of strings (of the same length as the number of
subsystems) naming the subsystems.
A vector of strings naming the exogenous inputs to the
final system. Each named input must be a different row
— however parenthesis can be used to denote vector valued
input names.
sysOutputs
A vector of strings defining the exogenous outputs of the
final system. These are defined in terms of input names or
subsystem outputs.
sysConnects A vector of strings (of the same length as the number of
subsystems) defining the input to each subsystem.
subsys1
Subsystems to be connected. The subsystems can be com-
binations of matrices and dynamic systems or combina-
tions of matrices and pdms.
.
.
.
“
Outputs: sys
The interconnected system, which can be either a dynamic
system, a pdm, or a matrix depending on the subsystems.
Description
sysic interconnects subsystems to form a single, larger system. The interconnection
specification strings and the actual subsystems are passed as arguments to sysic. In
this context, systems can be either Dynamic Systems or pdms. An interconnection
with both Dynamic Systems and pdms is not well defined and will generate an error.
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Matrices can also be included in the interconnection and are considered to be constant
gains when connected with Dynamic Systems and considered to be constant for all
domain values when connected with pdms.
sysic is able to connect linear systems together. For more complete interconnection and
simulation capabilities SystemBuild should be used.
Limitations
Only 20 subsystems can be interconnected with a single sysic call.
Example
# A standard unity gain negative feedback system
# is set up. To illustrate some of the capabilities
# an output noise and plant input disturbance are also
# added. The output signals are the plant output,
# controller effort and 10 x reference error.
p = 1/makepoly([1,0.1,1],"s")
c = makepoly([1,1],"s")/makepoly([1,10],"s")
snames = ["p";
"c"]
# name for subsystem 1
inputs = ["ref";
# the name is arbitrary
"dist(2)"]
outputs = ["p + dist(1)";
"c";
# this input is a dim 2 vector
# note reference to first dist input
"10*ref - 10*p"] # linear combinations can be used
conx = ["c + dist(2)"; # input to snames(1), i.e. "p"
"ref - p - dist(1)"] # input to "c"
clp = sysic(snames,inputs,outputs,conx,p,c)
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trsp
Syntax
[y,uint] = trsp(Sys,u,tfinal, ord,intstep)
Parameter List
Inputs:
Sys
Continuous dynamic system. The initial states are used in
the simulation.
PDM. Input signal.
u
tfinal
Final time in the simulation (optional). Default = max
time specified in u.
Keywords: ord
Scalar valued. Specifies order of interpolation for continu-
ous signals in the system. Options are 0 or 1. Default =
0.
intstep Scalar valued. Integration step size. If not supplied a
default will be calculated based on the system eigenvalues.
It will be rounded to make it divide into the discrete system
sample time by an integer value of at least 2.
Outputs:
y
PDM. Output signal.
uint
Interpolated version of u.
Description
Time domain simulation of a continuous system. This is conceptually the equivalent of:
y = Sys ∗ u.
This function allows the user more control over the parameters in the simulation.
The user can select an integration step (intstep) independently of the data spacing in
u. Furthermore, the spacing of u can be non-regular and u will be interpolated if
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necessary. If intstep is not specified the integration stepsize is determined from the
system eigenvalues and the minimum spacing in the input signal, u.
A zero or first order discrete equivalent of the systems can be specified. The standard
Xmath * operator uses only a zero order discretization. The first order simulation
actually uses a triangle hold equivalent. This is equivalent to connecting the samples
going into the hold function. Although the continuous time hold is noncausal the
discrete state-space system is causal. This is often more accurate at higher frequencies.
Examples
# We will use trsp to simulate the output of
# a second order system for various inputs
P = 400/makepoly([1,10,400],"s")
time = [0:0.25:2]
u = gstep(time,time,[0,1,0,-1,0,1,0,-1,0])
# Examine the standard Xmath simulation
y = P*u
gph1 = ctrlplot(y);
gph1 = plot(u,gph1,{marker=1,line=0,legend=["output";"input"]})?
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1.5
1
0.5
0
-0.5
output
input
-1
-1.5
0
0.5
1
1.5
2
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# Compare trsp calculation to standard
[ytrsp,uint] = trsp(P,u)
gph2 = ctrlplot(ytrsp);
gph2 = plot(y,gph2,{line style=4});
gph2 = plot(u,gph2,{line=0,marker=1});
gph2 = plot(uint,gph2,{line style=3,legend=["trsp calc.";...
"* calc.";"input";"interpolated input"],title=...
"Time response calculation comparisons"})?
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Time response calculation comparisons
1.5
1
0.5
0
-0.5
trsp calc.
* calc.
input
-1
interpolated input
-1.5
0
0.5
1
1.5
2
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# Now look at 1st order interpolation
[y1trsp,u1int] = trsp(P,u,{ord=1})
gph3 = ctrlplot([y1trsp,u1int]);
gph3 = plot(u,gph3,{line=0,marker=1,legend=...
["1st order interpolation";...
"interpolated input";"input data"],title=...
"Time response calculation comparisons"})?
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Time response calculation comparisons
1.5
1st order interpolation
interpolated input
input data
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
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truncate
Syntax
sysout = truncate(sysin,ord)
Parameter List
Inputs:
sysin
ord
Input Dynamic System
Order of the truncated system: sysout
Outputs: sysout Truncated output Dynamic System
Description
The function truncate is cross-licensed from the model reduction toolbox and has
slightly more capabilities than those described here. See the online help for further
details.
Consider a partitioning of the input Dynamic System, sysin, as follows.
A11 A12 B1
A21 A22 B2
sysin =
,
C1 C2
D
where A11 is of dimension ord × ord. The truncated output Dynamic System, sysout
is simply,
ꢀ
ꢁ
A11 B1
C1
sysin =
.
D
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Example
This example is identical to that described for sresidualize and compares the two
methods of model reduction.
# Create a five state system for reduction.
a = daug(-0.891334,[-1.20857,0.799042;-0.799042,-1.20857],...
-4.74685,-21.3013)
b = [0.0262569;-0.189601;-0.113729;0.211465;-0.538239]
c = [0.120725,-0.336942,0.397198,-0.700524,-1.02235]
d = 0
sys1 = system(a,b,c,d)
# Reduce to a 3 state system by residualization
# and truncation.
sysout1 = sresidualize(sys1,3)
sysout2 = truncate(sys1,3)
fHz = logspace(0.01,100,100)
sys1g = freq(sys1,fHz)
sysout1g = freq(sysout1,fHz)
sysout2g = freq(sysout2,fHz)
residerror = sys1g - sysout1g
truncerror = sys1g - sysout2g
gph1 = ctrlplot([sys1g,sysout1g,sysout2g,residerror,...
truncerror],{logmagplot});
gph1 = plot(gph1,{!grid,legend=["original system";...
"residualized system";"truncated system";...
"residualization error";"truncation error"]})?
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0.1
0.01
0.001
0.0001
1e-05
original system
residualized system
truncated system
residualization error
truncation error
0.01
0.1
1
10
100
Frequency
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See also
rifd, simtransform, sresidualize, orderstate modalstate.
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6.2 Xµ Subroutines and Utilities
Several subroutines may also be of interest to the user. These subroutines typically
perform self contained parts of a calculation. They may be of interest to those
developing new robust control algorithms or wishing to study the calculation details of
the algorithms given here. Beware of the fact that these subroutines may not contain
error checking.
These subroutines are included in alphabetical order and are cross-referenced by their
calling functions in the following list.
hinfsyn subroutines
hinfcalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
riccati eig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
riccati schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
mu subroutines
blkbal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
powermu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
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blkbal
Syntax
d = blkbal(M)
Description
Balances a square matrix assuming only scalar blocks. The Osborne method (growth
rate: n2) is used for large systems and the Perron method (growth rate: n3) for smaller
systems. The Perron method will exactly calculate mu for positive matrices.
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hinfcalc
Syntax
[X,Y,f,h,Ric fail,HX,HY,HXmin,HYmin] = ...
hinfcalc(p,nmeas,ncon,g,epr,{keywords})
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Parameter List
Inputs:
p
Generalized interconnection structure (Dynamic Sys-
tem)
nmeas
ncon
g
measurement vector dimension.
control vector dimension.
H
∞
norm of suboptimal controller to be calculated. Re-
ferred to in the literature as gamma.
Tolerance for determining when the Hamiltonian eigenval-
ues lie on the jω-axis.
epr
Keywords: schur solution real Schur decomposition for Riccati solution (default)
eig solution
eigendecomposition for Riccati solution.
Outputs:
X
Riccati solution
Riccati solution
status of solution:
Y
Ric fail
0
1
2
Solution found.
jω axis eigenvalues in Hamiltonian
Unequal number of +ve & -ve eigenvalues in
Hamiltonian. This represents a numerical failure
in the eigenvalue ordering.
3
Both of the above errors detected.
f
Intermediate calculation for scaling and normalization
Intermediate calculation for scaling and normalization
X Hamiltonian
h
HX
HY
Y Hamiltonian
HXmin
Minimum absolute value of the real part of the X Hamilto-
nian eigenvalues. In other words, how close to the jω axis
the eigenvalues lie.
HYmin
Minimum absolute value of the real part of the Y Hamil-
tonian eigenvalues.
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Description
Form and solve the Riccati equations for the H control problem. X and Y are the
∞
resulting Riccati solutions.
THIS FUNCTION IS INTENDED ONLY AS A SUBROUTINE CALLED BY THE
HINFSYN FUNCTION.
*** NO ERROR CHECKING ***
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powermu
Syntax
[lbnd,delta,errstat] = powermu(M,blk,rp,cp)
Description
Lower bound power algorithm based on the work of Andy Packard. The vector naming
roughly corresponds to that in his thesis.
*** NO ERROR CHECKING ***
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riccati eig
Syntax
[x1,x2,stat,Heig min] = riccati eig(H,epp)
Parameter List
Inputs:
H
Hamiltonian matrix.
epp
Tolerance for detecting proximity of eigenvalues to the jω
axis.
Outputs: x1,x2
stat
Basis vectors for stable subspace. See description below.
Status flag.
0
1
Stable subspace calculated.
Failure to decompose into stable and
unstable subspaces.
Heig min Minimum absolute value of the real part of the eigenvalues
of H.
Description
Solve the algebraic Riccati equation,
A0X + XA + XRX − Q = 0,
by an eigenvalue decomposition method. The Hamiltonian, H, contains the Riccati
equation variables in the matrix,
ꢀ
ꢁ
A
Q
R
−A0
H =
.
If H has no jω axis eigenvalues then there is an n dimensional (n = dim(A)) stable
subspace of H. The vector, [x1 x2] spans that stable subspace and, if x1 is invertible, the
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variable, X = x2x−1 1, is the stabilizing solution to the Riccati equation.
If H has jω axis eigenvalues then no stabilizing solution exists and the function returns a
failure status. If any eigenvalue of H is within epp of the jω axis it is considered to lie on
the jω axis and no solution is found. This may be due to numerical problems in finding
the eigenvalues of poorly conditioned problems even when a stabilizing solution exists.
See Also
Riccati, riccati schur
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riccati schur
Syntax
[x1,x2,stat,Heig min] = riccati schur(H,epp)
Parameter List
Inputs:
H
Hamiltonian matrix.
epp
Tolerance for detecting proximity of eigenvalues to the jω
axis.
Outputs: x1,x2
stat
Basis vectors for stable subspace. See description below.
Status flag.
0
1
Stable subspace calculated.
Failure to decompose into stable and
unstable subspaces.
Heig min Minimum absolute value of the real part of the eigenvalues
of H.
Description
Solve the algebraic Riccati equation,
A0X + XA + XRX − Q = 0,
by a real Schur decomposition method. The Hamiltonian, H, contains the Riccati
equation variables in the matrix,
ꢀ
ꢁ
A
Q
R
−A0
H =
.
If H has no jω axis eigenvalues then there is an n dimensional (n = dim(A)) stable
subspace of H. The vector, [x1 x2] spans that stable subspace and, if x1 is invertible, the
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variable, X = x2x−1 1, is the stabilizing solution to the Riccati equation.
If H has jω axis eigenvalues then no stabilizing solution exists and the function returns a
failure status. If any eigenvalue of H is within epp of the jω axis it is considered to lie on
the jω axis and no solution is found. This may be due to numerical problems in finding
the eigenvalues of poorly conditioned problems even when a stabilizing solution exists.
See Also
Riccati, riccati eig
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Appendices
A Translation Between Matlab µ-Tools and Xµ
This appendix outlines the functional equivalences between the Matlab µ-Tools and
Xmath Xµ. The objective is to provide a smooth transition for users moving from
µ-Tools to Xµ. We will assume that the reader is familiar with Matlab µ-Tools and the
general operation of Xmath. The intent is that the overall functional capabilities are the
same under either system and a prospective robust control designer chooses between
Matlab, Xmath, or future matrix languages, on other issues (cost, support, software
compatibility, etc.).
There are enough similarities between µ-Tools and Xµ that one can move from one to the
other without a great deal of additional learning. Most of the differences are syntactic.
There is not always a direct function for function match between the two systems.
The major differences are:
• The built-in data structures available with Xmath.
• The discrete-time system is available as an Xmath data object.
• Slightly different function names.
• The mu and musynfit functions handle the scaling and perturbation matrices in
matrix, rather than coded vector, form.
• The implementation of the D-K iteration is slightly different.
391
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The functionally equivalent commands will be listed, in each sub-section, for
convenience. A more detailed discussion is given to illustrate the more subtle differences
in the mu and D-K iteration aspects. More importantly, the Himat demo is available in
both µ-Tools and Xµ. For a fast start on moving between platforms, study these demos
side by side.
A.1 Data Objects
The underlying data structure in Matlab is the complex valued matrix, whereas in
Xmath there various other objects available: polynomials, transfer functions, parameter
dependent matrices, and dynamic systems.
system/Dynamic System Functions
The µ-Tools functions for creating and manipulating the systemmatrix have no Xµ
equivalent as the underlying Xmath operators can directly handle the dynamic system
and transfer function data objects.
Also residing within the Xmath state-space object is the initial state. This is used
primarily for time response calculations. It is debatable whether or not the initial state
is an intrinsic attribute of the system as one frequently changes it for simulation. It does
have the advantage of reducing the time response calculation to a simple multiplication
and it can easily be changed without accessing all of the other system variables.
The following table illustrates the equivalent functions, or indicates the data objects
which provide the same functionality.
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Description
µ-Tools Function Xmath/Xµ equivalent
form system
decompose system
form system
form system
decompose system
form system
random system
generate filters
fit transfer functions drawmag
pck
unpck
system
abcd
nd2sys
zp2sys
sys2pss
pss2sys
sysrand
mfilter
transfer function data objects
transfer function data objects
dynamic system data objects
dynamic system data objects
randsys
butterworth, chebyshev
fitsys, tfid
The µ-Tools drawmag function provides an interactive graphical interface. There is
currently no equivalent in Xµ. The equivalent underlying data fitting algorithm can be
found in the Xµ function fitsys.
varying/pdm Functions
The equivalent functions for construction and manipulation of pdm data objects are
shown below.
Description
µ-Tools Function Xmath/Xµ equivalent
form varying
break up varying vunpck
vpck
pdm
makematrix
get ivs.
join varying
sort by iv.
select by iv. value
select by index
scale iv.
getiv
domain
tackon
sortiv
xtract
xtracti
scliv
concatseg, insertseg, mergeseg
mergeseg
extractseg, indexlist, find
indexing by pdm(i)
domain, pdm
compare ivs.
random varying
indvcmp
varyrand
check
randpdm
As an aside, note that the sort function in Xmath sorts each column of each matrix in a
pdm, rather than sorting the domain.
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Subblocks: selecting input & outputs
In µ-Tools the function sel selects rows and columns from a varying matrix or inputs
and outputs from a system matrix. In Xmath these can be obtained by specifying row
and column indexes. More flexibility of selecting parts of a pdm can be obtained by
using the indexlist function.
Augmentation
Augmentation is the building of matrices from component pieces. The µ-Tools
commands which perform these functions are given in the table below. For pdms or
Dynamic Systems in Xmath these operations are generally performed identically to
the equivalent matrix operation.
Description
µ-Tools Function Xmath/Xµ equivalent
vertical augmentation
horizontal augmentation sbs
diagonal augmentation daug
abv
;
,
daug
Algebraic Operations
Similarly, algebraic operations on system or varying matrices in µ-Tools require a
dedicated function. In the Xmath case the usual matrix operation suffices.
Description
µ-Tools Function Xmath/Xµ equivalent
addition
subtraction
madd
msub
+
-
multiplication
system scaling
system scaling
system scaling
system inverse
transpose
mmult
mscl
sclin
sclout
minv
*
*
*
*
inv
’
*’
transp
conjugate transpose cjt
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Note that the transpose and conjugate transpose operators are defined differently for
Matlab and Xmath.
system/Dynamic System Functions
The following functions perform useful manipulations to, or information about, the state
of a system or Dynamic System.
Description
µ-Tools Function Xmath/Xµ equivalent
calculate poles
calculate zeros
display poles
spoles
szeros
rifd
poles
zeros
rifd
state similarity transform
reorder state
transform to modal format
zero order hold equivalent
statecc
reordsys
strans
samhld
simtransform
orderstate
modalstate
discretize
discretize
Prewarped bilinear transform tustin
varying/pdm Functions
A large number of varying matrix operations have been written in µ-Tools. These are
not required in the Xmath version as most of the functions operate on pdms as well as
matrices. The following table lists these functions for µ-Tools and Xmath. These are
only approximate functional equivalents — the more complicated functions differ in
some important respects.
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Description
µ-Tools Function Xmath/Xµ equivalent
peak norm
pkvnorm
vabs
norm, max
abs
diagonal
round
round
imag
real
conj
absolute value
diagonal matrix
round downwards
round upwards
imaginary part
real part
complex conjugate
norm
determinant
eigenvalues
vdiag
vfloor
vceil
vimag
vreal
vconj
vnorm
vdet
norm
det
eig
veig
inverse
left division
right division
pseudo-inverse
spectral radius
singular values
condition number
vinv or minv
vldiv
vrdiv
vpinv
vrho
inv
\
/
pinv
spectrad
svd
vsvd
vrcond
rcond or condition
schur decomposition vschur
schur
expm
interp or interpolate
pdm(vector)
matrix exponential
interpolation
decimation
vexpm
vinterp
vdcmate
vfft
FFT
fft
inverse FFT
spectral analysis
vifft
vspectrum
ifft
spectrum
Beware of syntactical differences here. One obvious example is the order in which the
eigenvalues and eigenvectors are returned from veig and eig. Note also the vrcond
returns the inverse of the condition number whereas condition returns the condition
number.
The µ-Tools function vebe performs element-by-element function operations on a
varying matrix. This has no counterpart in Xµ as the basic functions which operate on
each element of a matrix (e.g. sin, cos, abs) are also defined on pdms in Xmath. For the
same reason, the µ-Tools function veval has no Xµ counterpart: the Xmath execute
command can be used to the same effect.
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Miscellaneous Utilities
Several utilities are provided in µ-Tools. These are subroutines used by other µ-Tools
functions which may be of more general use.
Description
µ-Tools function Xmath/Xµ equivalent
complex random number
fit system to data
Eigenvalue based Riccati solution ric eig
Schur based Riccati solution ric schr
crand
fitsys
randpdm
fitsys
riccati eig
riccati schur
A.2 Matrix Information, Display and Plotting
Xmath provides matrix/data object size information via a variable window. There is no
Matlab equivalent for this functionality. Command window information can be
obtained via the following functions.
Description
matrix information minfo
list workspace whos
µ-Tools function Xmath/Xµ equivalent
check, size, is
who
Plotting of varying matrices is provided by the Matlab µ-Tools function vplot. As
pdms are a native data object in Xmath, the standard Xmath plot function will
correctly plot a pdm. Multiple calls will overlay the data, even if the domains differ. The
Xµ function ctrlplot is provided for more control specific plots: Bode, Nyquist,
Nichols, log magnitude, etc..
Both Matlab and Xmath allow interactive manipulation of the graphical data and
storage and retrieval of plots from the workspace or the underlying file system.
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A.3 System Response Functions
Creating Time Domain Signals
The Xmath pdm data object allows the creation of time domain signals via standard
and operators.
Description
µ-Tools function Xmath/Xµ equivalent
cosine waveform
sine waveform
stair-step waveform step tr
general waveform siggen
cos tr
sin tr
cos
sin
gstep
standard functions & operators
Time Responses
Xmath calculates time responses with the * operation. However, the Xµ function trsp
provides significantly more functionality in the continuous time case. The equivalences
are summarized below.
Description
µ-Tools function Xmath/Xµ equivalent
continuous time response trsp
trsp, *, deftimerange
*
sdtrsp
discrete time response
sampled data response
dtrsp
sdtrsp
Frequency Responses
The relevant functions are shown below.
Description
frequency response frsp
logarithmic vector logspace
µ-Tools function Xmath/Xµ equivalent
freq
logspace
The default frequency units in µ-Tools are radians/second whereas those in Xmath are
Hertz. This applies generically to all commands where the user specifies frequency
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information.
A.4 System Interconnection
Simple interconnection has already been outlined in the augmentation section above.
The more complicated interconnection functions are almost identical in µ-Tools and Xµ.
The only real difference is the calling syntax of sysic.
Description
µ-Tools function Xmath/Xµ equivalent
Redheffer (star) product starp
starp
sysic
system interconnection
sysic
A.5 Model Reduction
The model reduction functions in Xµ are:
Description
µ-Tools function Xmath/Xµ equivalent
residualization
state truncation
balanced realization
sresid
strunc
sysbal
sresidualize
truncate
balance
Hankel norm reduction hankmr
ophank
The following frequency weighted model reduction functions have no equivalent in the
current version of Xµ. It is intended to introduce this functionality in the next version.
The functions are: sdecomp, sfrwtbal, sfrwtbld, sncfbal and srelbal.
A.6 H2 and H Analysis and Synthesis
∞
The synthesis algorithms are identical and the syntax are similar.
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Description
µ-Tools function Xmath/Xµ equivalent
H2 norm calculation
h2norm
hinfnorm
h2syn
h2norm
hinfnorm
h2syn
H
norm calculation
∞
H2 controller synthesis
H
controller synthesis hinfsyn
hinfsyn
∞
The major syntactical difference is that the Xµ functions do not return the closed loop
system. This is easily calculated by a subsequent call to starp. The reason for this is
that the D-K iteration changes typically involve a differently weighted closed loop
system in subsequent operations. Having a separate calculation of the closed loop
reduces the potential for confusion.
The µ-Tools function hinffi has no Xmath/Xµ equivalent. This function calculates the
full information H controller. hinfsyn solves the more general problem and is of more
∞
practical use in controller applications.
A.7 Structured Singular Value (µ) Analysis and Synthesis
The issue of whether or not the frequency domain variable is radian/second or Hertz
arises here. Although this makes no difference to the calculation of µ is affects how the
D scales and ∆ perturbations are interpolated. µ-Tools assumes that the frequency scale
is radians/second. In Xµ the default assumption is Hertz. The reason for this is that
Hertz is the default output of the frequency response calculation freq. In all cases
where it makes a difference, the user can specify the keyword {!Hertz} to switch the
meaning of the domain.
Calculation of µ
There is a difference in the returned variable format for the Xµ function mu. The
Matlab function returns the D-scale and perturbations in coded vector form. The Xµ
mu function returns both the D scale and its inverse in matrix form. The relevant
functions are summarized below.
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Description
µ-Tools function Xmath/Xµ equivalent
structured singular value mu
mu
D scale decoding
unwrapd
not required
not required
blknorm
mkpert
perturbation decoding
block norm calculations
rational perturbation
random perturbations
unwrapp
blknorm
dypert
randel
randpert
In the Xµ mu function, only the default options of the µ-Tools mu calculations are
available. In other words, a power iteration, with several random restarts, is used for the
lower bound. The upper bound calculation uses an Osborne balancing method and
enhances this with the Perron vector method for problems with less than 10 blocks.
These methods have been found to be appropriate for the vast majority of practically
motivated problems.
New algorithms for these calculations are currently under development. The most
significant enhancement is the ability to calculate µ with respect to structures which
include real valued blocks. Because of the development effort in this direction, a wide
range of calculation options were not provided for the current Xµ mu function.
The scalar × identity block structure is not currently supported in the Xµ mu function.
It will be included in the revised version discussed above.
The D-K Iteration
There is a significant difference in the way that Xµ handles the D-K iteration. The D
scales are not incrementally factored into the previous iteration D-scales. Consider the
initial design interconnection structure to be ic. An H design will produce the first
∞
controller: k1, using a function call like the following.
k1 = hinfsyn(ic,nmeas,ncon,gamma)
The closed loop system, obtained via
g1 = starp(ic,k1)
is then analyzed with mu. The typical function call is:
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g1g = freq(g1,omega)
[mubnds,Dmagdata] = mu(g1g,blk)
From this, frequency domain D-scales are obtained, and a rational approximation is
obtained via musynfit.
[Dsys,Dinvsys] = musynfit(Dmagdata,blk,nmeas,nctrls,weight,g1g)
A difference between the Xµ and µ-Tools implementations of musynfit is that, in the
Xµ case, the inverse scaling system, Dinvsys, is generated by musynfit. Notice also
that the previous D-scale fit is not passed as an argument.
The new, D-scale weighted interconnection structure, ic2, is formed by,
ic2 = Dsys*ic*Dinvsys
A second design, k2, is now obtained with hinfsyn. Up to this point, the µ-Tools and
Xµ procedures are essentially the same.
The major difference is that the appropriate closed loop system is now formed with ic,
not ic2. In other words,
g2 = starp(ic,k2)
Note that ic and ic2 differ only by the D-scales and therefore µ(g2) would be the same
whether ic or ic2 were used for the closed loop system.
Now mu is used to analyze the frequency response ofg2, giving rise to a new set of
D-scales. Again, musynfit is used to fit state-space systems to these D-scales, giving
rise to D2sys and D2invsys. Now the next interconnection structure, ic3, is formed by
using these D-scales with the original interconnection. In command line form:
ic3 = D2sys*ic*D2invsys
Note that at each step, ic, is used to calculate the closed loop system, and also used to
calculate the next design interconnection structure.
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The advantage of this is that in order to restart, or reproduce, an iteration, one need
only save the previous controller. The µ-Tools approach requires saving the rational
approximation to the previous D-scales. The controller is a more applicable data object
to save and the saving of the previous D-scales depends on the quality of the rational
approximation. The disadvantage is that the upper bound in the next µ calculation
takes slightly longer as the closed-loop system does not have the numerical benefit of the
effects of the previous D-scales. The speed difference is likely to be insignificant in
practical problems.
Fitting D Scales
Rational transfer function fitting of magnitude data is required for the D-K iteration
and has been mentioned above. The relevant functions are summarized here.
Description
µ-Tools function Xmath/Xµ equivalent
fit D scale data
phase calculation
musynfit
genphase
musynfit
mkphase
transfer function fitting fitsys
fitsys, tfid
Note that the µ-Tools version of musynfit does not use the previous D scale magnitude
data. This has implications in the D-K iteration and is discussed in more detail above.
The alternative µ-Tools linear programming approach for D scale fitting, musynflp, is
not supported in Xµ. Similarly, the underlying linear programming approach to transfer
function fitting (µ-Tools functions fitmaglp, magdata) is not available in Xµ. The batch
functionality of the µ-Tools function, muftbtch, is available in the Xµ version of
musynfit.
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